TKK Dissertations 235Espoo 2010

ELECTROMAGNETIC AND MECHANICAL FINITE ELEMENT ANALYSIS OF END REGION OF LARGE-SIZED THREE-PHASE SQUIRREL-CAGE INDUCTION MACHINESDoctoral Dissertation

Ranran Lin林然然

Aalto UniversitySchool of Science and TechnologyFaculty of Electronics, Communications and AutomationDepartment of Electrical Engineering

TKK Dissertations 235Espoo 2010

ELECTROMAGNETIC AND MECHANICAL FINITE ELEMENT ANALYSIS OF END REGION OF LARGE-SIZED THREE-PHASE SQUIRREL-CAGE INDUCTION MACHINESDoctoral Dissertation

Ranran Lin林然然

Doctoral dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Faculty of Electronics, Communications and Automation for public examination and debate in Auditorium S4 at the Aalto University School of Science and Technology (Espoo, Finland) on the 17th of September 2010 at 12 noon.

Aalto UniversitySchool of Science and TechnologyFaculty of Electronics, Communications and AutomationDepartment of Electrical Engineering

Aalto-yliopistoTeknillinen korkeakouluElektroniikan, tietoliikenteen ja automaation tiedekuntaSähkötekniikan laitos

Distribution:Aalto UniversitySchool of Science and TechnologyFaculty of Electronics, Communications and AutomationDepartment of Electrical EngineeringP.O. Box 13000 (Otakaari 5)FI - 00076 AaltoFINLANDURL: http://sahkotekniikka.tkk.fi/en/Tel. +358-9-47001Fax +358-9-470 22991E-mail: ranran.lin@gmail.com

© 2010 Ranran Lin

ISBN 978-952-60-3285-6ISBN 978-952-60-3286-3 (PDF)ISSN 1795-2239ISSN 1795-4584 (PDF)URL: http://lib.tkk.fi/Diss/2010/isbn9789526032863/

TKK-DISS-2792

Picaset OyHelsinki 2010

ABSTRACT OF DOCTORAL DISSERTATION AALTO UNIVERSITYSCHOOL OF SCIENCE AND TECHNOLOGYP.O. BOX 11000, FI-00076 AALTOhttp://www.aalto.fi

Author Ranran Lin 林然然

Name of dissertation

Manuscript submitted 31 March 2010 Manuscript revised 28 June 2010

Date of defence 17 September 2010

Article dissertation (summary + original articles)Monograph

FacultyDepartmentField of researchOpponentSupervisorInstructor

Abstract

Keywords eddy currents, end winding, induction machine, leakage flux, leakage inductance, vibration

ISBN (printed) 978-952-60-3285-6

ISBN (PDF) 978-952-60-3286-3

Language English

ISSN (printed) 1795-2239

ISSN (PDF) 1795-4584

Number of pages 74 + 46

Publisher Department of Electrical Engineering, Aalto University

Print distribution Department of Electrical Engineering, Aalto University

The dissertation can be read at http://lib.tkk.fi/Diss/2010/isbn9789526032863/

Electromagnetic and Mechanical Finite Element Analysis of End Region of Large-Sized Three-PhaseSquirrel-Cage Induction Machines

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Faculty of Electronics, Communications and AutomationDepartment of Electrical EngineeringElectromechanicsAntonios KladasAntero Arkkio

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This piece of research is related to the design of large-sized radial flux electric machines. It aims at studyingthe end-region magnetic field under the steady-state operation in order for an analysis of some importantelectromagnetic and mechanical phenomena that require careful thought during the design of the end region.

The research work is focused on the end region of two large-sized three-phase squirrel-cage inductionmachines. In addition to the end-region magnetic field, the stator end-winding leakage inductance, the eddycurrents inside the end shield and the end frame, and the end-winding forces and vibrations are covered. The3-D time-harmonic finite element analysis of the above aspects, carried out with a commercial softwarepackage, forms the main part of the research work. The corresponding measurement, as a complement tothe finite element analysis, is mainly for validating the models and the numerical calculations.

A 3-D time-harmonic finite element analysis of the end-region magnetic field with suitable boundaryconditions can give an accurate distribution of the magnetic field. The stator end-winding leakage only affectsthe core ends; hence, the end-winding leakage inductance can be calculated from the correspondingmagnetic energy separated from the total magnetic energy. The eddy currents inside the end shield and theend frame do not affect the end-region magnetic field seriously, and it is feasible to consider them as surfacecurrents and to model them with the standard impedance boundary condition. The end-winding forces causevibrations and deformation, but the amplitude of vibration and the degree of deformation are small because ofthe end-winding bracing system. With the bracing system, the natural frequency of the most excitable mode ofthe end winding and its bracing system is also greatly raised, so resonances may be avoided.

Under the steady-state operation of large-sized induction machines, the end-region magnetic field does notbring about severely disadvantageous electromagnetic and mechanical phenomena in the end region.However, from the viewpoint of optimisation, the end frame can be shifted farther from the end winding foreliminating potential hot spots. In addition, fewer bracing parts can be fixed to the knuckle portion and thelower part of the involute portion of the end connections for improving the cooling of the end region.

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Preface

This piece of research began in May 2006. Before the end of 2009, the researchwork had been carried out at the Department of Electrical Engineering, HelsinkiUniversity of Technology. At the beginning of 2010, Helsinki University of Tech-nology, Helsinki School of Economics, and University of Art and Design Helsinkiwere merged into Aalto University; hence, since then, the research work has beenconducted at the Department of Electrical Engineering, Aalto University.

First of all, I am indebted to my supervisor, Prof Antero Arkkio, who guided methroughout the whole of my research work. Prof Arkkio proposed many invaluablesuggestions and corrected many of my mistakes during the research work. I havelearnt quite a lot from the discussions with him.

I am most grateful to Prof Asko Niemenmaa, the head of the department, for hissupport and encouragement. I would like to express my gratitude to Emeritus ProfTapani Jokinen and Dr Anouar Belahcen for the discussions with them, and to DrJarmo Perho, the former laboratory manager, for the selection of my research topic.Prof Jorma Kyyra, a vice rector of Aalto University, is specially thanked for hiswarm-hearted help. Mr Ari Haavisto, an operations engineer of the department,deserves my gratitude for helping me with a lot of daily work as well as somemeasurement. Mr Ilkka Hanhivaara, the laboratory technician of the department,is thanked for helping me with the preparation of some measurement. Ms MarikaSchroder, a former secretary of the department, and Ms Anja Meuronen and MsTerhi Arvela, secretaries of the department, are also thanked for their nice help. Inaddition, my thanks go to all of my colleagues, with whom I had so many usefuldiscussions and so much fun.

Dr Antti Laiho, who is working in VTT Technical Research Centre of Finland, de-serves my grateful thanks for his help in vibration measurement. Dr Janne Roivainenand Dr Timo Holopainen, who are working in ABB Corporation in Helsinki, Finland,are also thanked for their invaluable comments on one of my papers.

I thank my Chinese colleagues working in the department, Dr Xiaozhi Gao, DrXiaolei Wang, and Mr Zengcai Qu, as well as some former Chinese visiting scholarsin the department, Dr Wei Chen from Tianjin University, and Mr Gang Liu fromJixi University.

The technical support of COMSOL Group is appreciated, and in particular, I amgrateful to Mr Pasi Marttila, Mr Sakari Lukkarinen, Dr Erik Danielsson, and MsAnna Dzougoutov.

I am much obliged to Prof Stefan Kulig from TU Dortmund University in Germanyand Prof Erich Schmidt from Vienna University of Technology in Austria for their

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invaluable comments on the manuscript of this doctoral thesis.

Graduate School in Electrical Energy Engineering in Finland is thanked for mycourse study. Moreover, Finnish Foundation for Technology Promotion (Tekni-ikan edistamissaatio) and Research Foundation of Helsinki University of Technology(Teknillisen korkeakoulun tukisaatio) are thanked for the financial support. Specialthanks go to China Scholarship Council for the award given to me.

Last but not least, I would like to express my sincere gratitude to my grandmother,Ms Jingfang Li, my parents, Mr Dezhong Lin and Ms Aixiao Li, and my wife, MsXuemei Qiu, for their support, care, and love. They play an important role in mylife, study, and work.

Espoo, Finland, July 2010

Ranran Lin

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Contents

Preface 5

Contents 7

List of Publications 9

Author’s contribution 11

List of Abbreviations 13

List of Symbols 15

1 Introduction 191.1 Background to Research . . . . . . . . . . . . . . . . . . . . . . . . 191.2 Aim of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3 Scope of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 Scientific Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 231.5 Structure of Compendium . . . . . . . . . . . . . . . . . . . . . . . 24

2 Review of Relevant Literature 252.1 End-Region Magnetic Field . . . . . . . . . . . . . . . . . . . . . . 252.2 Eddy Currents in End Region and Core Ends . . . . . . . . . . . . 282.3 Stator End-Winding Inductance . . . . . . . . . . . . . . . . . . . . 292.4 End-Winding Forces and Vibrations . . . . . . . . . . . . . . . . . . 312.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Basic Introduction to Finite Element Analysis 363.1 Finite Element Analysis of End-Region Magnetic Field . . . . . . . 363.2 Finite Element Analysis of End-Winding Vibrations . . . . . . . . . 38

4 Electromagnetic Analysis of End Region 414.1 End-Region Magnetic Field . . . . . . . . . . . . . . . . . . . . . . 414.2 Stator End-Winding Inductance . . . . . . . . . . . . . . . . . . . . 454.3 Eddy Currents inside End Shield and End Frame . . . . . . . . . . 48

5 Mechanical Analysis of End Region 535.1 End-Winding Magnetic Forces . . . . . . . . . . . . . . . . . . . . . 535.2 End-Winding Mechanical Vibrations . . . . . . . . . . . . . . . . . 55

6 Conclusions and Discussions 606.1 Main Results of Research . . . . . . . . . . . . . . . . . . . . . . . . 606.2 Scientific Significance of Research . . . . . . . . . . . . . . . . . . . 616.3 Accuracy of Research Methods . . . . . . . . . . . . . . . . . . . . . 62

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Appendix A Main Parameters of Electric Machines

Errata

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List of Publications

This doctoral thesis consists of two parts: a compendium, and a collection of relevantpublications. The publications are listed below and are referred to in the text bytheir Roman numerals.

I Lin, R., Haavisto, A., and Arkkio, A. (2009). Validation of a time-harmonicnumerical model for solving magnetic field in end region of a radial-fluxmachine. IEEE T Magn, 45(12):5360–5367.

II Lin, R. and Arkkio, A. (2009). Calculation and analysis of stator end-winding leakage inductance of an induction machine. IEEE T Magn,45(4):2009–2014.

III Lin, R., Haavisto, A., and Arkkio, A. (2010). Analysis of eddy-currentloss in end shield and frame of a large induction machine. IEEE T Magn,46(3):942–948.

IV Lin, R. and Arkkio, A. (2008). 3-D finite element analysis of magneticforces on stator end-windings of an induction machine. IEEE T Magn,44(11):4045–4048.

V Lin, R., Laiho, A. N., Haavisto, A., and Arkkio, A. (2010). End-windingvibrations caused by steady-state magnetic forces in an induction machine.IEEE T Magn, 46(7):2665–2674.

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Author’s contribution

The author of this doctoral thesis carried out most of the research work indepen-dently. He is the first-named author of the above publications, so he is responsiblefor most of the contents of the publications. The co-authors of the publications car-ried out the measurement related to the research work reported in the publicationswith the first-named author, gave the first-named author constructive ideas for theresearch work reported in the publications, and made invaluable comments on thecontents of the publications.

The following is a summary of the first-named author’s and the co-authors’ con-tribution to each of the above publications, which has been recognised by all theco-authors.

• Publication I was fully written by the first-named author. Haavisto, A. andthe first-named author performed the corresponding measurement of the elec-tromagnetic field together. Arkkio, A. gave the first-named author construc-tive ideas for the research work reported in the publication and made invalu-able comments on the contents of the publication.

• Publication II was fully written by the first-named author. Arkkio, A. gavethe first-named author constructive ideas for the research work reported in thepublication and made invaluable comments on the contents of the publication.

• Publication III was fully written by the first-named author. Haavisto, A.and the first-named author performed the corresponding measurement ofthe electromagnetic field together. Arkkio, A. gave the first-named authorconstructive ideas for the research work reported in the publication and madeinvaluable comments on the contents of the publication.

• Publication IV was fully written by the first-named author. Arkkio, A. gavethe first-named author constructive ideas for the research work reported in thepublication and made invaluable comments on the contents of the publication.

• Publication V was fully written by the first-named author. Laiho, A. N.and the first-named author performed the modal testing and the operatingdeflection shape measurement of the end-winding together. Laiho, A. N.also performed the data processing relating to the modal testing and themeasurement. Haavisto, A. and the first-named author performed the cor-responding measurement of the electromagnetic field together. Laiho, A. N.and Arkkio, A. both gave the first-named author constructive ideas for theresearch work reported in the publication and made invaluable comments onthe contents of the publication.

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List of Abbreviations

BVP boundary value problemCMP correlated mode pairDFT discrete Fourier transformDOF degree of freedomEMF electromotive forceFDM finite difference methodFEA finite element analysisFEM finite element methodFRF frequency response functionLTI linear time-invariantMAC modal assurance criterionODS operating deflection shapePM permanent-magnetSIBC standard impedance boundary conditionVPI vacuum pressure impregnationrms root-mean-square

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List of Symbols

A magnetic vector potential [Wb/m]Ar, Aφ, Az r-, φ-, and z-component of A [Wb/m]B magnetic induction [T]C viscous damping matrix [N·s/m]Cjk entry of C [N·s/m]D electric flux density [C/m2]D Rayleigh’s dissipation function [J/s]E electric field strength [V/m]Eact induced EMF of search coil in active region [V]Eend induced EMF of search coil in end region [V]Ek kinetic energy [J]Es strain energy [J]Ep potential energy [J]F column vector of external force [N]Few end-winding force [N]Fexc column vector of excitation force [N]Fi force on segment i [N]Fi,r, Fi,φ, Fi,z r-, φ-, and z-component of Fi [N]Fi,r,∼, Fi,φ,∼, Fi,z,∼ double-frequency component of Fi,r, Fi,φ, and Fi,z [N]Fj external force associated with qj [N]H magnetic field strength [A/m]J conduction current density [A/m2]Js source current density [A/m2]Jsur surface current density [A/m]K stiffness matrix [N/m]Kjk entry of K [N/m]Lstr

σ,end stator end-winding inductance per phase [H]M mass matrix [kg]Mjk entry of M [kg]N number of generalised coordinatesPFt eddy-current loss [W]PφFt eddy-current loss per unit circumferential length [W/m]Qj generalised force associated with qj [N]R str resistance of stator phase winding [Ω]S surface area [m2]T electric vector potential [A/m]Tz z-component of T [A/m]T1 period of fundamental frequency [s]V electric scalar potential [V]V volume [m3]Vi volume of segment i [m3]W reg

m,act magnetic energy in active region [J]

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W srcm,act magnetic energy associated with coil sides [J]

W regm,end magnetic energy in end region [J]

W srcm,end magnetic energy associated with end connections [J]

Wm,tot total magnetic energy [J]Zs surface impedance [Ω]cstl specific heat capacity of steel [J/(kg·K)]fi force density of segment i [N/m3]fi,r, fi,φ, fi,z r-, φ-, and z-component of fi [N/m3]i indexistr current of stator phase winding [A]j indexj imaginary unitk indexkFe stacking factor of corel strFe stator core length [m]m number of phasesn normal unit vectorp number of pole pairspd dissipated power density [W/m3]q column vector of generalised coordinateqj, qk generalised coordinates j and kr radial coordinates slipt time [s]u column vector of displacement [m]ustr voltage of stator phase winding [V]z axial coordinateΓ surface of integration∆t incremental time [s]∆ϑ incremental temperature [K]Ω domain of integrationϑi,r0, ϑi,φ0, ϑi,z0 phase angle of Fi,r,∼, Fi,φ,∼, and Fi,z,∼ [rad]µ permeability [H/m]ν tensor reluctivity [m/H]νFe,z axial reluctivity of core [m/H]νlam reluctivity of lamination [m/H]ν0 reluctivity of vacuum [m/H]ϱstl mass density of steel [kg/m3]σ tensor conductivity [S/m]σ conductivity [S/m]ϕ magnetic scalar potential [A]φ circumferential coordinateψcalc column vector of calculated mode shapeψexp column vector of experimental mode shape

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ψstract flux linkage of stator phase winding in active region [Wb]

ω angular frequency [rad/s]ωs angular frequency of stator field [rad/s]

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1 Introduction

This chapter forms an introduction to this compendium. The background to thispiece of research is explained at the beginning, followed by the aim and the scope.The scientific contributions are briefly summarised at the end of this chapter.

1.1 Background to Research

In conventional ac rotating electric machines, such as synchronous machines andinduction machines, the stator core and the rotor core are usually stacked with lam-inations made from non-oriented electrical steel sheets. The magnetic flux passingthrough the air gap is mainly in the radial direction, so these kinds of electric ma-chines are referred to as radial flux electric machines. Figure 1.1 shows a diagramof a large-sized radial flux electric machine.

Figure 1.1: A diagram of half of the drive end of a large-sized radial flux electricmachine (squirrel-cage induction machine).

Under the steady-state operation of radial flux electric machines, about 95% of thetotal magnetic flux is in the active region, and only around 5% in the end region.Therefore, their operating characteristics can be predicted relatively well even if theend-region magnetic field is totally omitted. However, under the transient operation,e.g. the starting of 3-phase induction motors, the extremely large starting currentcauses deep saturation of the laminated cores, which greatly increases the proportionof the end-region magnetic flux to the total magnetic flux. Under the circumstances,

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the end-region magnetic field becomes relatively important for predicting the op-erating characteristics of these electric machines. In addition, in some large-sizedradial flux electric machines, e.g. turbogenerators installed at power stations, evenunder the steady-state operation, the end-region magnetic field may cause somedisadvantageous electromagnetic and mechanical phenomena, which might have amajor effect on their operating characteristics. As a result, more attention must bepaid to the end region of these large-sized electric machines.

When large-sized synchronous machines or induction machines run under a load,the end-region magnetic field comprises mainly four components. First, the mostimportant one is the stator end-winding leakage, which is caused by the currentflowing in the end connections1 of the stator coils, as shown in Figure 1.1, since theend connections stretch deep into the end region. Second, the current flowing inthe end portion of the field winding of wound field synchronous machines, and thecurrent flowing in the rotor end winding of wound rotor induction machines or inthe rotor end cage2 of squirrel-cage induction machines cause leakage flux in the endregion of these electric machines. Third, the end-region magnetic field induces eddycurrents in the surrounding metal parts as well as in the core ends; conversely, thoseeddy currents may affect the end-region magnetic field. Finally, the air-gap fringingflux, as a part of the air-gap magnetic flux, is also distributed in the end region.

In general, the aforementioned eddy currents can give rise to resistive loss, andthe loss can generate heat and might cause hot spots in some of the surroundingparts. The time-varying end-region magnetic field can cause magnetic forces on theend connections of the stator coils, which may excite mechanical vibrations thatgive rise to looseness of the end-winding bracing system, damage to the enamelon the surface of the end connections, and looseness of the stator coils especiallyin stators having a non-global vacuum pressure impregnation (VPI) process (Stoneet al., 2009); conversely, the vibrations might slightly affect the end-region magneticfield. Moreover, the above heat and forces can cause stresses and strains in theend connections as well as in the parts surrounding them. Therefore, some of thesephenomena can affect one another, and their main interactions are shown in Figure1.2.

Since they become more serious under the transient operation, some of the abovephenomena must be taken into consideration during the design of large-sized electricmachines. On the other hand, the design of electric machines normally requires acalculation of the electromagnetic field, and the most important parameter associ-ated with the end-region magnetic field is the stator end-winding leakage inductance.Its accurate value is particularly important because an accurate value can greatlyraise the accuracy of both analytical models based on lumped parameter equiva-

1In this compendium, the definition of the term “end connections” is from reference (Gross,2007), and the term “end connections” means the portion of a coil outside the slots.

2In this compendium, the term “rotor end cage” means the portion of a rotor cage outside therotor slots, i.e. the end rings and the end bars.

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Figure 1.2: The interactions of the main phenomena in the end region of large-sizedradial flux electric machines.

lent circuits and 2-D cross-sectional numerical models based on coupled field–circuitequations.

1.2 Aim of Research

This piece of research aims at calculating the end-region magnetic field in large-sizedradial flux electric machines in order for an estimation of the stator end-windingleakage inductance and a detailed analysis of some important electromagnetic andmechanical phenomena arising in the end region, i.e. the eddy currents, the magneticforces, the stresses, and the mechanical vibrations.

1.3 Scope of Research

Solving the 3-D end-region magnetic field in large-sized radial flux electric machinesis at the core of this piece of research because the study of the above electromagneticand mechanical phenomena is mainly dependent upon the solution to the magneticfield. With the aid of increasingly advanced computers, numerical methods havebeen widely applied to the calculation of the magnetic field in radial flux electricmachines for several decades, but nowadays their application is still concentrated onthe 2-D numerical analysis of the magnetic field in the active region. However, thispiece of research is focused on the 3-D numerical analysis of the end-region magneticfield as well as the numerical analysis of the aforementioned eddy currents, magnetic

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forces, stresses, and mechanical vibrations. The numerical analysis performed in thispiece of research is based on the finite element method (FEM).

As the end-region magnetic field in large-sized radial flux electric machines is moreimportant than that in small-sized ones, this piece of research places great emphasison large-sized electric machines. Turbogenerators generally have a large size and alarge capacity, but it is hard to study them in 3-D space with the FEM because ofthe large number of degrees of freedom (DOFs) in the course of solving their 3-Dmodels. It is almost impossible to take any measurements of a turbogenerator inthe laboratory for validating corresponding numerical calculations. As a result, thispiece of research only covers large-sized 3-phase induction machines. In general,large-sized synchronous machines and induction machines both have similar form-wound multi-turn stator coils, as shown in Figure 1.3, and the main difference liesin the structure of the end portion of the rotor, which is normally not as importantas the stator end winding.

Figure 1.3: Half of a form-wound stator coil in the two-layer diamond winding ofa radial flux electric machine.

For the sake of economy, electromechanical devices, e.g. rotating electric machines,are generally designed in such a way that the electrical steel sheets of the cores areinto saturation during certain moments in each supply cycle (Williamson and Ralph,1982). Although the saturation of the cores and the rotation of the rotor are usuallytaken into account in the 2-D finite element analysis (FEA) of the magnetic fieldin the active region, to consider them in the 3-D FEA of the end-region magneticfield is not easy, on account of the extremely long computation time. On the otherhand, unlike the air-gap magnetic flux, the end-region magnetic flux forms closedloops mainly in the air and other non-magnetic substances, which means that thesaturation is not a dominant factor. Consequently, in this piece of research, all 3-Dcalculations relating to the end-region magnetic field are based on time-harmonicFEAs, but a few 2-D calculations are based on time-discretised FEAs. By takingsome necessary measurements as validations, the results obtained from the time-harmonic FEAs can be considered reliable.

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1.4 Scientific Contributions

The following is a summary of the scientific contributions of this piece of research.

1. A linear model solved in a time-harmonic FEA is capable of estimating the3-D end-region magnetic field in a large-sized induction machine accuratelywhen the anisotropic effect of the laminated cores is considered and suitableboundary conditions are applied.

2. The proposed method of estimating the magnetic energy associated with thestator end-winding leakage in a large-sized induction machine is applicablesince the stator end-winding leakage cannot penetrate the middle portion ofthe stator core. The stator end-winding inductance, which is used in the2-D coupled field–circuit numerical models and analytical lumped parametermodels, can be calculated from the corresponding magnetic energy.

3. It is feasible to model the eddy currents inside the end frame3 and the endshield of a large-sized induction machine with the standard impedance bound-ary condition (SIBC). The method of estimating the eddy-current loss bymeasuring the temperature rise is also applicable. The eddy-current lossonly accounts for a small portion of the total losses of an induction machine.

4. In a large-sized induction machine, the radial end-winding force density inthe knuckle portion of the end connections is larger than that in the otherportions. The end-winding forces on each of the end connections belongingto the same phase belt are somewhat different.

5. An undamped linear time-invariant (LTI) mechanical model is capable ofobtaining the modal data of the end winding and its bracing system and ofstudying the end-winding vibrations of a large-sized induction machine whencorrect mechanical properties are given. The amplitude of vibration is smallunder the steady state, even in the nose portion, and the natural frequencyof the most excitable mode is much higher than the excitation frequency.

The above five scientific contributions can be found in the corresponding publicationsin this thesis, as listed in Table 1.1 below.

Table 1.1: Scientific Contributions and Corresponding Publications.

Scientific contribution 1 2 3 4 5Corresponding publication I II III IV, V V

3In this compendium, the term “end frame”means the portion of a stator frame that lies in theend region.

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1.5 Structure of Compendium

This compendium is organised according to the following structure.

• Chapter 1 forms an introduction to this compendium. The background to thispiece of research, the aim, and the scope are explained. The correspondingscientific contributions are elaborated as well.

• Chapter 2 is intended as a review of the literature on the end-region magneticfield in large-sized radial flux electric machines as well as the correspondingeddy currents, magnetic forces, stresses, mechanical vibrations, etc.

• Chapter 3 outlines the FEA of the end-region magnetic field and the end-winding vibrations performed in this piece of research. The basic theory ofelectromagnetic fields and mechanical vibrations is briefly given as well.

• Chapter 4 forms a summary of the electromagnetic analysis of the end regionin large-sized squirrel-cage induction machines, including an analysis of theend-region magnetic field, a calculation of the stator end-winding inductance,and an analysis of the eddy currents inside the end shield and the end frame.

• Chapter 5 forms a summary of the mechanical analysis of the end regionin large-sized squirrel-cage induction machines, including an analysis of theend-winding forces and an analysis of the end-winding vibrations.

• Chapter 6 summarises the main results and discusses the scientific significanceof this piece of research. Some issues relating to the accuracy of the researchmethods are also discussed.

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2 Review of Relevant Literature

This chapter is intended as a review of the relevant literature on the end-regionmagnetic field in large-sized radial flux electric machines and related electromagneticand mechanical phenomena. The review falls into roughly four categories. The firstis focused on the end-region magnetic field; the second discusses the eddy currentsand the corresponding eddy-current loss; the third is about the stator end-windingleakage inductance, and in the fourth category, the end-winding forces and the end-winding vibrations are covered. In general, none of the above four categories istotally independent, so there is some overlap among the categories.

2.1 End-Region Magnetic Field

Before numerical methods were put into use, analytical methods were the only meansof calculating the end-region magnetic field in large-sized radial flux electric ma-chines, and they are still in use nowadays. For instance, in references (Chow et al.,1982) and (Frei-Spreiter and Reichert, 1998), the end-region magnetic field was con-sidered to be caused by two sources: the end connections of the stator coils andthe magnetisation sources, e.g. the magnetic field in the rotor shaft (Frei-Spreiterand Reichert, 1998). The computation of both parts caused by the two sources wascompleted by analytical integral equations, but the eddy currents in the end regionas well as in the stator core ends were not taken into consideration. In addition,in the analytical calculations, the end connections of the stator coils were usuallyreplaced by a number of infinitely thin segments along their centrelines, e.g. in ref-erence (Ban et al., 2005). In order for the inclusion of the effect of the stator core,the image method was used, and the images of the end connections were substitutedfor the stator core. For instance, Ban et al. (2005) and Schramm and Gerling (2005)illustrated a detailed procedure for the use of the image method.

Numerical methods like the FEM have been used in the study of the electromag-netic field in large-sized radial flux electric machines for several decades since theycan deal with arbitrary shapes, arbitrary boundary conditions, and complicatedor distributed sources (Salon, 1990). On account of the complex structure of theend region, a quasi-3-D method was used for simplifying the study, which meansthat the end-region magnetic field was considered to be a sinusoidal travelling wavepropagating in the circumferential direction (Ho and Fu, 1998). Therefore, only thechanges within the radial–axial plane required careful consideration (Ho and Fu,1998). As a matter of fact, the spatial distribution of the end-region magnetic fieldis not simply sinusoidal in the circumferential direction by reason of the harmonics.However, to solve the end-region magnetic field in 3-D space requires plenty of com-putational resources and perhaps a great deal of computation time, in particular,

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when time-discretised nonlinear FEAs are carried out.

Weiss and Stephens (1981) completed a 3-D magnetostatic analysis of the end-regionmagnetic field in a turbogenerator. The magnetic field was solved in such a waythat it was decomposed into two components: one was rotational and caused by thecurrent in the end connections, and the other was irrotational everywhere. The twocomponents were solved from the magnetic vector potential and the magnetic scalarpotential, respectively. The authors also took account of the effect of the slots andthe teeth on the circumferential permeability of the core. Plantive et al. (1996) alsoconducted a magnetostatic analysis. They solved the end-region magnetic field inan 820-MVA generator by using the two-potential method4.

An analysis of the end-region magnetic field in a 27-kVA synchronous generatorwas conducted by Richard et al. (1997), who used the two-potential method. Theauthors obtained a distribution of the magnetic induction above the nose portionof the end connections. They also found that the copper screen had a major effecton the magnitude and the distribution of the end-region magnetic field. In anotheranalysis, in order to evaluate the leakage flux in the end region as well as in thestator core ends of a 1000-MW turbogenerator under different power factors, Fujitaet al. (2000) proposed a relatively fast approach. They solved the magnetic field onlyunder an open circuit and a short circuit by the two-potential method. By usingphasor diagrams, they were able to evaluate the magnetic field under any powerfactor, both inductive and capacitive. In their calculation, the use of Biot-Savart’slaw kept the source currents excluded from the finite element mesh. In addition,as the involute portion of the end connections of form-wound coils is usually laidupon the surface of a fictitious cone with a certain conical angle (Holmberg, 1998), Liet al. (2005) solved the end-region magnetic field in a turbogenerator under differentconical angles by using the two-potential method. In their study, the conical anglewas varied from 30 to 31.5, but the authors found that there was little influenceon the distribution and the magnitude of the end-region magnetic field.

Sikora et al. (1982, 1986) solved the end-region magnetic field in an induction mo-tor by using the FEM based on the T –ϕ formulation, where T and ϕ denote theelectric vector potential and the magnetic scalar potential, respectively. In theirearlier study reported in reference (Sikora et al., 1982), the current density in theend connections was supposed to have an axial and a circumferential component.The authors calculated the magnetic field under two different boundary conditionsapplied to the stator core, the frame, etc., i.e. an infinitely permeable boundary con-dition (µ→ +∞) and an infinitely conductive one (σ → +∞), where µ and σ denotethe permeability and the conductivity, respectively. Reference (Sikora et al., 1986)is a continuation of the earlier one. The authors improved their model by takingaccount of all three components of the current density in the end connections.

4In this compendium, the term “the two-potential method” means the method in which thetotal and the reduced magnetic scalar potential are used.

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The nonlinear 3-D calculation of the end-region magnetic field has not been widelyconducted yet on account of its extremely long computation time. However, Noven-der and Muller (1983) completed a nonlinear calculation of the end-region magneticfield in a 214-MVA turbogenerator by the finite difference method (FDM), in whichsingle-valued B–H curves were used in the magnetic materials. The computationtimes of CPU for the no-load and the full-load calculation were around 61 h and75 h, respectively.

Jack and Mecrow (1986, 1987) published two papers on the investigation into theend-region magnetic field in a 660-MW turbogenerator as well as its effect on thestator core. They used a quasi-3-D method and solved the magnetic field by usingthe FEM based on the T –ϕ formulation. Nonlinear single-valued B–H curves wereused in the ferromagnetic materials, and the eddy currents inside the stator coreends were considered. The authors mainly analysed the radial variations of the axialmagnetic induction, i.e. from the tooth tip to the core back, in different positionsof the stator core ends.

In large-sized radial flux electric machines equipped with form-wound multi-turnstator coils, the distribution of the current density over the cross-section of theconductors is non-uniform, and there are some circulating currents flowing in theconductors. Tang et al. (1990) analysed the end-region magnetic field in a 300-MWturbogenerator as well as the circulating currents in the coils by the FEM basedon the A–V formulation, where A and V denote the magnetic vector potentialand the electric scalar potential, respectively. The authors pointed out that it wasthe circumferential and the radial component of the end-region magnetic field thatcaused the circulating currents.

Wen et al. (1994) analysed the end-region magnetic field in a 35-MW turbogenera-tor under a 3-phase short circuit by using a quasi-3-D method based on the A–Vformulation. The authors found that the eddy currents inside the retaining ringwere important and not negligible during the transient calculation of the end-regionmagnetic field. Yao et al. (2006) conducted an analysis of the end-region magneticfield in a 1056-MVA turbogenerator with the FEM, in which the reduced magneticvector potential was used. The modelling of the complicated source currents wasavoided in the finite element mesh, and the magnetic field caused by the sourcecurrents was calculated from Biot-Savart’s law instead. The topic of the end-regionmagnetic field in large-sized turbogenerators was also studied by Liang et al. (2003,2008) and Yao et al. (2008), and the A–V formulation was used in all the abovethree studies. Among them, Liang et al. (2003) dealt mainly with the magnetic fieldin the rotor end region.

A comparison of different methods of calculating the end-region magnetic field wasdrawn by Lazarns et al. (2009). From the viewpoint of the structural optimisationof electric machines, the comparison indicated that a method based on 2-D FEAs inwhich an axiperiodic distribution of the currents was supposed possessed acceptable

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accuracy and modest computational requirements, by comparison with analyticalmethods, the image method, and methods based on 3-D FEAs. Then the methodwas applied to the geometric optimisation of the magnetic shields of a permanent-magnet (PM) machine for reducing the end-winding forces.

2.2 Eddy Currents in End Region and Core Ends

The end-region magnetic field in large-sized radial flux electric machines can induceeddy currents in the surrounding metal parts. The eddy currents inside the lam-inations of the core ends are also partly related to the end-region magnetic field.In general, it is quite difficult or impossible to study those eddy currents with an-alytical methods because of their complicated distribution; hence, they are mainlyanalysed with numerical methods.

Jack and Mecrow (1987) and Mecrow et al. (1989) jointly analysed the eddy currentscaused by the axial flux in the stator core ends of turbogenerators with the FEMbased on the T –ϕ formulation. In reference (Jack and Mecrow, 1987), a quasi-3-D method was used and the authors found that the eddy currents were mainlyconcentrated around the end of the Pistoye slots on account of the slits. They alsoestimated the corresponding loss density in various positions of the end packets.In reference (Mecrow et al., 1989), the authors completed a thorough study of theaxial flux causing eddy currents in the stator core ends of turbogenerators. Theyevaluated the importance of different sources causing the axial flux. Various factorsrelated to the end-region magnetic field were also discussed, such as the steppedcore ends and the change in the rotor core length. In particular, how the effect ofthese factors impinged upon the axial flux and the corresponding loss was discussed.

Khan et al. (1990) published a paper dealing with the eddy-current loss and thetemperature caused by the axial flux in the stator core ends of a 1000-MW turbo-generator. They used a quasi-3-D method but took the nonlinear magnetic circuitinto account. The authors mainly analysed the distribution of the axial magneticinduction in the end packets in both the axial and the radial direction as well asthe distribution of the loss density in the axial direction. It was found that thedouble-slit stator teeth could reduce more eddy currents than the single-slit statorteeth. Silva et al. (1995a) especially analysed the effect of the slitting of the statorteeth in a 300-MVA generator. The authors did not take account of the saturationof the core. The eddy currents induced by the axial flux in the core ends were mod-elled by the SIBC, which was applied to the surface of the stator core ends. Withthe single-slit teeth, a remarkable drop in the eddy-current loss was found, but aslight increase in the loss in the stator core back was observed. In another studyperformed by Silva et al. (1996), the authors analysed the same eddy-current lossagain with the FEM based on the T –ϕ formulation. In comparison with reference(Silva et al., 1995a), part of the stator core with radial cooling ducts was included,

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and the laminated core was treated as a homogeneous, anisotropic solid core. Theymainly illustrated the distribution of the eddy currents as well as the distributionof the loss in the core ends.

Schmidt et al. (2005) published a paper on the effect of the harmonics of the end-region magnetic field on the eddy currents in the stator clamping system of a 450-MVA hydroelectric generator. Both a nonlinear time-discretised and a linear time-harmonic FEA based on the A–V formulation were conducted. They found thatthe calculations of the loss obtained from both of the cases were quite close in theclamping plate and the clamping fingers when just the fundamental of the excitationcurrent was considered. However, when the 3rd harmonic was added, the calcula-tions of the loss in the clamping fingers were still close in both of the cases, butthere was a big difference between the calculations in the clamping plate.

Then Yao et al. (2006) analysed the distribution of the eddy currents in the fingerplate and the clamping plate of a 1056-MVA turbogenerator by using the reducedmagnetic vector potential and Biot-Savart’s law. Moreover, in another paper writ-ten by Yao et al. (2008), the eddy currents in the clamping plate of a 1000-MWturbogenerator were analysed with the A–V formulation. The authors found thatthe eddy currents were mainly distributed around the inner surface of the clampingplate. The corresponding eddy-current loss was also calculated under different op-erating conditions, such as no-load and full-load. In reference (Liang et al., 2008),by using the A–V formulation, the authors briefly analysed the distribution of theeddy currents in the pressing plate of a 150-MW turbogenerator under no-load andunder full-load, respectively.

In the above studies, the laminated stator core was modelled as a solid core withelectrical and magnetic anisotropy — in other words, the laminations were not reallymodelled by reason of the extremely small thickness of both themselves and theirinsulation layers. However, Yamazaki et al. (2008) completed a calculation of theeddy currents in the stator core ends of a 250-MVA turbogenerator by using theFEM based on the A–V formulation, and the laminations were really modelled inthe finite element mesh used in their calculation.

2.3 Stator End-Winding Inductance

The leakage flux in the end region of radial flux electric machines includes mainly theleakage coming from the stator parts, e.g. the stator end winding, and the leakagecoming from the rotor parts, e.g. the rotor end cage, but the most important leakagecomponent is the stator end-winding leakage, which is usually represented by statorend-winding inductance.

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Ban et al. (2005) completed an analytical calculation of the stator end-winding in-ductance of a 247-MVA turbogenerator by using a method based on Neumann’sintegral. They took full account of the 3-D structure of the end connections ofthe stator coils, especially the involute portion. The effect of the stator core wasconsidered with the image method. By comparing their calculations with the mea-surements, they found that the calculations were close to the measurements whenthe relative permeability of the stator core was supposed to be zero — in otherwords, the stator core exhibited a highly impermeable effect. Another similar an-alytical calculation was performed by Schramm and Gerling (2005). The statorend-winding inductance of a switched reluctance machine was calculated with ananalytical method quite similar to the one reported in reference (Ban et al., 2005).Additionally, Hsieh et al. (2007) carried out another analytical calculation. In theircalculation, the end connections of the stator coils were considered to be semicircu-lar. The magnetic induction was calculated from the magnetic vector potential, andthen the end-winding inductance was determined from the flux linkage inside theend connections. The authors found their method fast and accurate in comparisonwith other analytical methods, but they did not discuss how effective the methodwould be when the real geometric shape of the end connections was considered.

Apart from the above analytical computations, most computations are based onnumerical methods. In reference (Taieb Brahimi et al., 1993), the calculation ofthe stator end-winding inductance was based on the idea that the inductance ofthe whole stator winding varied as a linear function of the stator core length. Intheir calculation, the end region and the stator core ends of each machine wereanalysed from a 3-D model, but the middle portion of the core was analysed from a2-D model. The end-winding inductance was determined by linear extrapolation; inother words, in terms of the relationship between the inductance of the whole statorwinding and the core length, the end-winding inductance was extrapolated when thecore length reached zero. Another numerical calculation was performed by Chiveret al. (2008). An idea, similar to the one reported in reference (Taieb Brahimi et al.,1993), was adopted in the calculation. Unlike Taieb Brahimi et al., the authorscalculated the inductance of a portion of the stator winding lying in the activeregion of each machine by using a 2-D model, and then they estimated the end-winding inductance of each machine by subtracting the above inductance from theinductance of the whole stator winding, which was calculated from a 3-D model.

Cox et al. (2008) also studied the topic of the end-winding inductance, but theyfocused on the concentrated modular winding used in PM machines. The sameprocedure as in reference (Chiver et al., 2008) was used, i.e. a combination of a 2-Dand a 3-D numerical model. In terms of the numerical calculations, they developedan analytical equation in order to conveniently estimate the end-winding inductanceof the concentrated modular winding of a certain type of machines. Liang andChen (2001) completed a calculation of the end-winding inductance of a 600-MWturbogenerator by using a quasi-3-D method. The authors investigated the accuracyof the calculations under different conditions, i.e. with and without the modelling

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of the eddy currents in the end region, and with different rotor core lengths but afixed stator core length.

In reference (de Weerdt and Belmans, 1995), not only the end-winding inductancebut also other types of leakage inductance of an induction machine were thoroughlydiscussed. The authors first used a current-driven 2-D axisymmetric model andfound that the end-ring inductance was heavily dependent upon the boundary con-ditions of the model. Next, in their current-driven 3-D model, the end-windinginductance, the end-ring inductance, and the mutual inductance between the endring and the end winding were analysed, and they all depended on the slip. For in-stance, there was an increase in the mutual inductance when the slip increased, buta drop in both the end-ring inductance and the end-winding inductance appeared.Furthermore, in another paper written by de Weerdt et al. (1997), the authors anal-ysed the importance of different end-region parameters of an induction machine,such as the end-winding inductance and the end-ring inductance, under differentoperating conditions, i.e. under no-load, full-load, and locked-rotor operation.

Different types of leakage inductance of large-sized electric machines were also cal-culated by Arshad et al. (2005), such as the end-winding inductance and the slotleakage inductance. The authors mainly analysed the accuracy of the calculationsof the leakage inductance on the basis of different numerical models, such as 2-Dmodels and 3-D models with and without radial cooling ducts.

2.4 End-Winding Forces and Vibrations

In radial flux electric machines running normally or abnormally, the time-varyingend-winding forces cause mechanical vibrations of the end winding and its bracingsystem. The stresses and the strains in the end winding as well as in its bracingsystem are also partly related to the end-winding forces. The above aspects areclosely correlated, so they are often considered together in the relevant literature.

Scott et al. (1981) calculated the steady-state end-winding forces of turbogenerators.In their analysis, the end-region magnetic field was supposed to vary sinusoidallyin the circumferential direction and was solved with the FDM. The effect of thepower factor on the end-winding forces was also analysed. In addition, as a contin-uation of reference (Scott et al., 1981), reference (Salon et al., 1983) was focusedon the transient end-winding forces. In that paper, the spatial distribution of theforces under different kinds of malfunctions was shown, e.g. a 3-phase short circuitfrom full-load, and it was found that the transient forces tended to spread the endconnections out.

Khan et al. (1989) also studied the transient end-winding forces in turbogenerators.A 1000-MW turbogenerator was analysed, with the current in the field winding as

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well as in the damper winding considered. The spatial distribution of the forcesunder a 3-phase short circuit from full-load was also shown, which is similar to thedistribution reported in reference (Salon et al., 1983). Then, in a paper writtenby Wen et al. (1996), the transient end-winding forces in a 35-MW turbogeneratorunder other kinds of malfunctions, such as a line-to-line short circuit and a line-to-neutral short circuit, were investigated.

Kim et al. (2005) completed an analysis of the transient end-winding forces duringthe starting of a high-voltage induction motor. The authors compared the end-winding forces among different phase belts and then analysed the reliability of theinsulation layer of the end connections in terms of the stresses there. Liu andHjarne (2007) performed a numerical calculation of the end-winding forces in a 635-MVA turbogenerator under full-load and a 3-phase short circuit from no-load. Theauthors calculated the force vectors by introducing a local coordinate system, whichwas defined in different positions of the end connections. They found that the forcesalways tended to spread the end connections out, whether under full-load or underthe short circuit. It was also revealed that the forces on the two layers of the endwinding could be attraction or repulsion. Additionally, in an analysis performedby Stancheva and Iatcheva (2009), in addition to the end-winding forces, the forcesexerted on the stator core ends of a 200-MW turbogenerator were calculated withthe Maxwell stress tensors.

Among the studies dealing with the end-winding vibrations, Patel and Butler (1983)analysed the relationship between the power factor and the end-winding vibrationsof large-sized synchronous generators. The vibration patterns measured in variouspositions of the stator end winding were shown. By analysing the effect of the powerfactor and the core vibrations on the end-winding vibrations, the authors found thatthe end-winding vibrations were significantly strengthened under capacitive powerfactors and that there was a strong coupling between the core vibrations and theend-winding vibrations. Lambrecht and Berger (1983) systematically studied themechanical properties of the integrated end-winding bracing system of turbogener-ators. The authors measured the end-winding vibrations under three conditions:unexcited motor-driven operation to determine the relationship between the end-winding vibrations and the rotor speed; no-load operation to determine the end-winding vibrations caused by the core vibrations, and short-circuit operation todetermine the end-winding vibrations caused by the forces. On the basis of themeasurements, the effect of the vibration components with different frequencies onthe amplitude of vibration was investigated.

Additionally, Leger and Szylowicz (1997) published a paper on the modelling ofthe end-winding vibrations of two turbogenerators. In their paper, a proportionaldamping model (Rayleigh damping model) was used in the viscous damping matrixof the governing equation. They also outlined the effect of the deterioration of thebracing system, e.g. the looseness of the wedges in the nose portion. In a paperwritten by Zhang et al. (1998), the authors summarised the consequences of the

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end-winding vibrations in turbogenerators. The method used for the measurementand the method used for reinforcing the end-winding vibrations were explained.

Drubel et al. (2000) analysed the end-winding deformation in different turbogener-ators under full-load and a 3-phase short circuit. The authors considered differentkinds of design of the end-winding bracing system and found that the temperature-dependent stiffness of the tapes had a major effect on both the natural frequenciesand the degree of deformation. Furthermore, they revealed that the resonant fre-quencies of the end winding were related to the change in the temperature.

Another important aspect is to monitor the long-term end-winding vibrations inturbogenerators. In reference (Demcko et al., 2007), a complete procedure for moni-toring the end-winding vibrations based on fibre-optic accelerometers was explainedin detail. In recent years, artificial intelligence algorithms, e.g. neural networks,have often been employed in order for the analysis of the variables to be monitored.For instance, van Wyk and Hoffman (2002) used a feedforward network in orderto identify the long-term trend of the vibrations from the short-term end-windingfluctuations, and an auto-associative network in order to reduce the number of inputvariables.

Among the studies dealing with the stresses and the strains in the end winding,Ohtaguro et al. (1980) completed an analysis of the mechanical properties of thestator end winding in a 900-kW induction motor during its starting. At first, theauthors found that the strains in the insulation layer of the end connections weredependent upon the number of spacers and their positions. Next, they ran a bendingtest for the coils with different kinds of insulation layers and concluded that thenonlinear relationship between the deflection and the bending moment must betaken into consideration in the course of calculating the deflections and the strains.Additionally, in reference (Merkhouf et al., 2003), the von Mises stresses in the end-winding bracing system of a 110-MVA synchronous generator under the transientoperation were calculated. The authors found that the stresses calculated at twopossible resonant frequencies were much lower than the corresponding yield stress.

In a paper written by Nagano et al. (2008a), the authors analysed the forces on theend winding with connection rings and those on the end winding without connectionrings, and found that the effect of the current in the connection rings on the forceswas negligible. They also performed a modal analysis of the above two end windings,and concluded that the difference between the two calculated modal models wasquite small. Furthermore, in their another study reported in reference (Naganoet al., 2008b), the method of analysing the end-winding vibrations is similar tothe one reported in reference (Lambrecht and Berger, 1983) — in other words, therotation of the rotor and the core vibrations were both taken into account. Theauthors also analysed the stresses in the insulation layer of the end connections,which were caused by the end-winding forces and the thermal expansion. It wasfound that the largest stress appeared in the vicinity of the knuckle portion.

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2.5 Summary

Many studies relating to the end region of large-sized radial flux electric machines,especially turbogenerators, have been carried out so far, and most of them arefocused on the aforementioned aspects: the magnetic field, the eddy currents, theend-winding inductance, and the forces and vibrations.

First, an analysis of the end-region magnetic field is basic to an analysis of the otheraspects. Analytical and numerical methods both have been adopted in the analysis,but the FEM is the most common method currently. With the FEM, the magneticfield has been solved with the two-potential method, or the A–V formulation, orthe T –ϕ formulation. In fact, the A–V formulation is suitable for the 2-D FEAof the cross-sectional model of radial flux electric machines in which the magneticfield and the winding circuit supplied from a voltage source are coupled and solvedtogether at the same time. However, the T –ϕ formulation is quite effective whenmost of the analysed domain is free of current (Nakata et al., 1988), e.g. the endregion of electric machines, but more attention should be paid to the cancellationproblem, which occurs in the iron part with high permeability (Nakata et al., 1990).Nowadays, most commercial FEA software packages use the A–V formulation.

Second, the eddy currents related to the end-region magnetic field have been studiedby numerical methods, especially the FEM, on the basis of either the A–V or theT –ϕ formulation. Most of the analyses are focused on the eddy currents in themetal parts of the end region, e.g. the clamping plate, as well as in the laminationsof the core ends of turbogenerators. Actually, with a suitable coordinate system,the T –ϕ formulation is effective in modelling the 2-D eddy currents inside the planeof the laminations because T can have only one axial component, Tz, which isperpendicular to the plane of the laminations. However, the eddy currents insidethe end shield and the end frame of large-sized electric machines have not beenanalysed yet.

Third, in the analytical calculation of the stator end-winding inductance, a com-bination of Neumann’s integral and the image method is normally used in orderfor the inclusion of the effect of the stator core. In fact, most of the calculationsare based on the analysis of the magnetic energy, which is usually calculated withthe FEM. The idea of those numerical calculations is that the inductance of thetotal stator winding, including the end-winding inductance, is a linear function ofthe stator core length, so the end-winding inductance can be derived from linearextrapolation.

Fourth, the end-winding forces in turbogenerators have been analysed as Lorentzforces, mostly with numerical methods, especially under the transient operation,e.g. under a 3-phase short circuit. In the course of the analysis of the correspondingvibrations, the amplitude of vibration, the resonant frequencies, and the stresses

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in the end connections are the main issues. The effect of the core vibrations andthe power factor on the end-winding vibrations has also been discussed. However,only the end-winding vibrations in turbogenerators are covered, and the vibrationsappearing in large-sized induction machines have not been reported in detail.

Although all the aforementioned aspects of the end region have been studied, someissues have still not been covered, such as the eddy currents inside the end shieldand the end frame as well as their effect on the end-region magnetic field, etc. Inparticular, the electromagnetic and the mechanical phenomena in the end regionof large-sized induction machines have not been studied so much, and thereforeanalysing these phenomena in large-sized induction machines is the main focus inthis piece of research.

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3 Basic Introduction to Finite Element Analysis

This chapter outlines the FEA of the end-region magnetic field and the end-windingmechanical vibrations performed in this piece of research. The basic theory ofelectromagnetic fields and mechanical vibrations is briefly given as well. Detailedimplementation of the FEA is not stated owing to the use of a commercial softwarepackage.

3.1 Finite Element Analysis of End-Region Magnetic Field

The 3-D, time-varying electromagnetic field in rotating electric machines can besolved through Maxwell’s equations. The differential form of the relevant Maxwell’sequations is

∇×E = −∂B∂t

(3.1)

∇×H = J +∂D

∂t(3.2)

where E denotes the electric field strength, B the magnetic induction, t the time, Hthe magnetic field strength, J the conduction current density, andD the electric fluxdensity. The displacement current density ∂D/∂t in (3.2) can be omitted becausethe following relation holds under quasi-static conditions:

∂D

∂t≪ J . (3.3)

In addition, related constitutive relations are

H = ν ·B (3.4)

J = σ ·E (3.5)

where ν and σ denote the tensor reluctivity and the tensor conductivity, respectively.

With the introduction of the magnetic vector potential A defined by B, i.e.

B = ∇×A, (3.6)

and the introduction of the electric scalar potential V , (3.1) is transformed into

E = −∂A∂t

−∇V. (3.7)

By the above (3.3)–(3.7), (3.2) is transformed into

∇× [ν· (∇×A)] +σ·∂A∂t

+ σ·∇V= 0. (3.8)

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In eddy-current problems, (3.8) is valid in the region in which eddy currents areconsidered. Moreover, when there are some known source currents in the region inwhich there are no eddy currents, the source current density Js can be added in(3.8), i.e.

∇× [ν· (∇×A)] +σ·∂A∂t

+ σ·∇V − Js = 0. (3.9)

In (3.9), there are either source currents or eddy currents in any current-carryingregion, but not both. Actually, it is the governing equation of the A–V formulationcommonly used in the 3-D FEA of the electromagnetic field in electric machinesas the current in the winding is normally considered a known source current. Inaddition, in the region in which eddy currents are considered, the continuity ofcurrent should be satisfied:

∇ ·(σ·∂A

∂t+ σ·∇V

)= 0. (3.10)

When the field quantities are supposed to vary sinusoidally in the time domain, thetime dependence of (3.9) and (3.10) can be eliminated through a use of complexvectors and phasors. Under the circumstances, (3.9) and (3.10) become

∇× [ν· (∇×A)] + jωσ·A+ σ·∇V − J s= 0 (3.11)

∇ · (jωσ·A+ σ·∇V ) = 0 (3.12)

where a line under a symbol denotes the complex vector or the phasor of the fieldquantity symbolised by the symbol, ω the angular frequency, and j the imaginaryunit.

In fact, the above forms a theoretical part of the A–V formulation. The A–Vformulation has been widely adopted in many software packages relating to theFEA of electromagnetic fields, e.g. COMSOL MultiphysicsTM.

From a mathematical point of view, solving the electromagnetic field in electricmachines is equivalent to seeking a solution to a complicated boundary value problem(BVP) described by its differential equation, e.g. (3.9). Numerical methods havebeen widely used for solving such complicated BVPs because of their feasibility andaccuracy. In this piece of research, the analysis of the 3-D end-region magneticfield in large-sized squirrel-cage induction machines was based on the FEM, and thecorresponding FEA was completed with COMSOL MultiphysicsTM, on the basis ofthe A–V formulation.

When the magnetic field in radial flux electric machines is solved from A by 2-Dcross-sectional models in which the geometric structure and the material propertiesare independent of the axial position, A has just one axial component, Az. Thepotential component Az is a scalar quantity, so nodal elements, in which scalarshape functions are associated with the nodes of finite elements, are used for seeking

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a piecewise planar function to approximate the distribution of Az. With nodalelements, the values of Az at the free nodes of finite elements are solved, and Az iscontinuous across the interface between any two neighbouring finite elements.

However, the end-region magnetic field must be considered in 3-D space, so Ahas all its r-, φ-, and z-component, Ar, Aφ, and Az, where r, φ, and z denotethe radial, the circumferential, and the axial coordinate, respectively. Obviously,what needs to be solved is a vector quantity. Although it is still possible to usenodal elements to solve A, e.g. in reference (Keskinen, 1992), edge elements, inwhich vector shape functions are associated with the edges of finite elements, arewidely used in the FEA of 3-D magnetic fields instead of nodal elements. Withedge elements, the values of the tangential component of A along the free edges offinite elements are solved. They can also ensure the continuity of the tangential fieldcomponent across the interface between any two neighbouring finite elements andcan leave the normal field component discontinuous (Kameari, 1990; Webb, 1993;Mur, 1994). Therefore, it is possible to solve a 3-D magnetic field directly from itsfield quantities, e.g. H . As pointed out by Jin (1993), edge elements can overcomesome shortcomings of nodal elements: the occurrence of spurious solutions which isattributed to a lack of enforcement of the divergence condition; the inconvenience ofapplying boundary conditions to material interfaces as well as conducting surfaces,and the difficulty in treating conducting and dielectric edges and corners due to thefield singularities associated with these structures. A detailed explanation of edgeelements lies beyond the scope of this compendium, but it can be found in relevantbooks, e.g. in references (Jin, 1993) and (Volakis et al., 1998).

In COMSOL MultiphysicsTM, edge elements and nodal elements are used for dis-cretising the distribution of the magnetic vector potential A and the electric scalarpotential V , respectively. In addition, the discretisation with the FEM is based onthe weak formulation of the weighted residual formulation of the differential equa-tion, and the weighted residual formulation is completed with Galerkin’s method. Inall the FEAs of the end-region magnetic field performed in this piece of research, be-cause of the limited computational resources, first-order tetrahedral and first-orderprismatic edge elements were used for discretising the 3-D distribution of A in theend region and in the active region of the induction machines, respectively, but nogauge condition was applied to A.

3.2 Finite Element Analysis of End-Winding Vibrations

There is no difference between the mechanical vibrations in electric machines andthe vibrations of other dynamic systems. In general, there are two approachesto studying the vibrations of a dynamic system: a force approach and an energyapproach. The force approach is based on Newton’s second law of motion and theenergy approach is based on the energy conservation principle. The energy approach

39

is suitable for approximate computer methods, e.g. the FEM (Doyle, 2004).

The energy approach is related to the principle of virtual work. On the basis of thisapproach, the governing equation of the motion of a dynamic system can be derivedfrom Lagrange’s equation of motion, which uses generalised coordinates that areindependent of each other and equal in number to the DOFs of the system. Fora linear dynamic system with translational motion described by its N generalisedcoordinates, the Lagrange’s equation of motion given in reference (Doyle, 2004) is

d

dt

(∂Ek

∂qj

)− ∂Ek

∂qj+

∂

∂qj(Es + Ep) −Qj = 0, j = 1, · · · , N, (3.13)

where a dot over a symbol denotes the first derivative of the field quantity symbolisedby the symbol with respect to time, j the index, qj generalised coordinate j, Ek thekinetic energy of the system, Es the strain energy of the system, Ep the potentialenergy of the system, and Qj the generalised force associated with qj. Among theabove physical quantities, Ep is related to the conservative forces, but Qj is relatedto the nonconservative forces.

As fully described in reference (Doyle, 2004), when the motion of the dynamic systemis small, with a Taylor series expansion about its equilibrium position“0”, the kineticenergy can be obtained as

Ek ≈1

2

N∑j=1

N∑k=1

Mjkqj qk (3.14)

where k denotes the index, qk generalised coordinate k, and

Mjk =∂2Ek

∂qj∂qk

∣∣∣∣∣0

. (3.15)

Similarly, the strain energy can be obtained as

Es ≈1

2

N∑j=1

N∑k=1

Kjkqjqk (3.16)

where

Kjk =∂2Es

∂qj∂qk

∣∣∣∣∣0

, (3.17)

and the potential energy can be obtained as

Ep = −N∑j=1

Fjqj (3.18)

where

Fj = − ∂Ep

∂qj

∣∣∣∣∣0

(3.19)

40

where Fj denotes the external force associated with generalised coordinate qj.

When the nonconservative forces of the system are considered as viscous forces, Qj

can be derived from a potential function called Rayleigh’s dissipation function, D,as

Qj = −∂D∂qj

(3.20)

where

D ≈ 1

2

N∑j=1

N∑k=1

Cjkqj qk (3.21)

where

Cjk =∂2D

∂qj∂qk

∣∣∣∣∣0

. (3.22)

With (3.14), (3.16), (3.18), (3.20), and (3.21), the Lagrange’s equation of motioncan be organised in a form of matrix as

Mq +Cq +Kq = F (3.23)

where M denotes the mass matrix whose entries are shown in (3.15), C the viscousdamping matrix whose entries are shown in (3.22), K the stiffness matrix whoseentries are shown in (3.17), q a column vector of the generalised coordinates, F acolumn vector of the external forces whose entries are shown in (3.19), and two dotsover a symbol denote the second derivative of the field quantity symbolised by thesymbol with respect to time.

As a matter of fact, the above forms a part of the basic theory of the mechanicalvibrations of dynamic systems. Lagrange’s equation of motion is based on gener-alised coordinates, but (3.23) can be applied to displacement analyses as well pro-viding that the coordinates are measured from a fixed reference system, as pointedout by Vierck (1979). In addition, (3.23) is a governing equation used in manysoftware packages relating to the FEA of mechanical vibrations, e.g. COMSOLMultiphysicsTM, which was used in this piece of research in order for the analysis ofthe end-winding vibrations.

In general, the end-winding vibrations in electric machines should be analysed in3-D space because of the 3-D structure of the end winding. In other words, thedisplacement caused by the vibrations has all its r-, φ-, and z-component. In COM-SOL MultiphysicsTM, nodal elements are used for discretising the distribution of thedisplacement. In all the FEAs of the end-winding vibrations performed in this pieceof research, second-order tetrahedral nodal elements were used for the discretisation.

41

4 Electromagnetic Analysis of End Region

This chapter outlines the electromagnetic FEA of the end region of two large-sized 3-phase squirrel-cage induction machines. The FEA covers three aspects: an analysisof the end-region magnetic field, a calculation of the stator end-winding inductance,and an analysis of the eddy currents inside the end shield and the end frame. Adetailed account of the above three aspects is not provided in this chapter, but itcan be found in Publications I–III.

4.1 End-Region Magnetic Field

The end-region magnetic field in a 3-phase, 4-pole, 2.24-MW squirrel-cage inductionmachine was analysed. The main parameters of the machine are listed in AppendixA as Machine I. The rotor was removed so that the motor could be tested in thelaboratory. The machine has a two-layer diamond winding consisting of insulatedform-wound multi-turn coils. A diamond winding was selected in the FEA becauseits model was easier to built with commercial software packages than other kindsof windings, e.g. random windings commonly used in low-voltage electric machines(Sadarangani, 2000). In fact, diamond windings are commonly used in radial fluxelectric machines with a power range of 1 MW and above, and a voltage range of 1kV – 30 kV (Sadarangani, 2000). Their detailed description can be found in manybooks, such as references (Liwschitz-Garik and Whipple, 1946), (Holmberg, 1998),and (Sadarangani, 2000).

The model of the end connections was built by the measurements of the 3-D coor-dinates in different positions of a certain end connection, and each end connectionin the model was made up of 22 connective segments. The multi-turn coils weremodelled as single-turn solid coils. The laminated core was modelled as a homoge-neous solid core with magnetic and electrical anisotropy. As in references (Bastosand Quichaud, 1985) and (Silva et al., 1995b), from the corresponding equivalentreluctance, its axial reluctivity was determined from

νFe,z = νlamkFe + ν0 (1 − kFe) (4.1)

where νFe,z denotes the axial reluctivity of the core, νlam and ν0 the reluctivity ofthe laminations and that of vacuum, respectively, and kFe the stacking factor of thecore. The axial conductivity of the solid core was considered to be zero because ofthe insulation layers on the surface of the laminations. In fact, the laminated corewas stacked with non-oriented electrical steel sheets instead of grain-oriented ones,but the non-oriented ones still had higher permeability in their rolling direction. Inthe process of manufacturing the core, the laminations were rotated with respectto each other so that the magnetic properties of the core could be as isotropic as

42

possible in the plane of the laminations. Therefore, in this piece of research, themagnetic and the electrical properties of the solid core were considered isotropicin all directions perpendicular to the axial direction. In addition, because of theperiodicity and the symmetry of the magnetic field, only one pole at one end of themachine was modelled, but the model did not include the rotor. Figure 4.1 showsthe finite element mesh of the end region and that of a certain end connection.

(a) (b)

Figure 4.1: The finite element mesh of the model of a 3-phase, 4-pole, 2.24-MWsquirrel-cage induction machine. (a) The finite element mesh of the end region. (b)The finite element mesh of a certain end connection.

The current in the end winding of the model was supplied from a current source. Thenonlinearity of the ferromagnetic materials was not taken into consideration becausea nonlinear calculation had to be carried out by a time-discretised FEA requiringextremely long computation time. Therefore, a time-harmonic FEA was performed.As the time dependence of the field quantities was eliminated in the time-harmonicFEA, the reluctivity used in the approximation should be independent of time. Aspointed out by Arkkio (1987), an approximation of the sinusoidal time-variationwas only reasonable when root-mean-square (rms) values were calculated underthe steady state, so it was normally a good choice to define the reluctivity of theferromagnetic materials as a function of the rms value of the magnetic induction.However, it was not possible to define the reluctivity as mentioned above in thecommercial software package used. It was also difficult to give different values of thereluctivity in the ferromagnetic parts having different degrees of saturation, e.g. thelaminations, in the commercial software package. Therefore, a 2-D time-discretisedFEA based on a cross-sectional model of the same machine was completed in orderfor the examination of the average reluctivity over the plane of the laminations.

43

From the calculations, the unsaturated reluctivity was around 1.74 × 10−4ν0, andthere was an increase in the reluctivity as the magnetic field was strengthened. Atthe rated operating point, it was around 2.64 × 10−4ν0. Because of the skin effect,the actual reluctivity of the core could be even larger. As a result, a common valueof the reluctivity, 0.001ν0, was used in the solid core. The use of this value definitelycaused an error in the calculation of the magnetic field inside the core, but the end-region magnetic field was hardly affected in terms of the measurements. This valuewas used in the end shield and the end frame as well.

The governing equation of the FEA was based on (3.11), but the term σ·∇V wasdropped because of the limited computational resources. As a matter of fact, in theeddy-current region, σ·∇V in (3.11) always reduced the flow of the eddy currents,and therefore, from the viewpoint of eddy currents, the omission of σ·∇V repre-sented a worst case scenario. For instance, in Section 4.3, the calculations based ona simple model show that the eddy-current loss inside the end shield and the endframe becomes around 10% smaller when σ·∇V is included. The term jωσ·A in(3.11) is the dominant part of the eddy currents. As a result, the actual governingequation of the FEA is

∇× [ν· (∇×A)] + jωσ·A− J s= 0. (4.2)

The boundary conditions of the model were analysed, in particular, the boundaryconditions dealing with the end shield and the end frame made from constructionsteel. The skin depth of those parts was around 1 mm at 50 Hz when its reluctivitywas 0.001ν0. According to Bastos and Sadowski (2003), it was necessary to discre-tise the eddy-current region into small finite elements in order for good precision.However, if the finite element mesh there had been generated accurately, a largenumber of finite elements would have been generated and the computer would nothave been able to solve such a problem. Therefore, those parts had to be replacedby suitable boundary conditions.

In the FEA, three kinds of boundary conditions were applied to the inner surfaceof those parts, as shown in Figure 4.2. The corresponding three cases are explainedbelow.

• Case 1: A homogeneous Dirichlet boundary condition was applied as

A× n = 0 (4.3)

where n denotes a normal unit vector. According to Morisue (1988), theboundary condition meant that the normal component of B vanished withinthe inner surface of those parts.

• Case 2: A homogeneous Neumann boundary condition was applied as

∇×A× n = 0. (4.4)

44

Figure 4.2: Three different kinds of boundary conditions applied to the innersurface of the end shield and the end frame. (Source: Publication I.)

It meant that the tangential component of B vanished within the inner sur-face of those parts.

• Case 3: In order that the eddy currents within the inner surface of thoseparts could be modelled, the SIBC was applied as

n×E = Zsn× (n×H) (4.5)

where Zs denotes the surface impedance.

In sum, in terms of the results obtained from the measurement performed in thefollowing three cases: underload current (500 A), full-load current (832 A), andoverload current (1000 A), the nonlinearity caused by the saturation is not distinctin the end region in all the three cases, by comparison with the nonlinearity inthe active region. In the case of full-load current, the discrete Fourier transform(DFT) of the time-varying induced electromotive force (EMF) of the search coil inthe end region indicates that a 5th and a 7th time harmonic arise in the end-regionmagnetic field, in addition to the fundamental, but they are quite weak. The resultsalso show that the axial magnetic induction is the dominant component within thesurface of the end shield but the radial magnetic induction is the dominant onewithin the surface of the end frame. In addition, the induced EMF of the searchcoils, the voltage of the stator phase winding, and the magnetic induction in the endregion were measured respectively. In comparison with the measurements, the time-harmonic FEA based on all the above boundary conditions can obtain acceptablecalculations of the end-region magnetic field, but the calculations based on theboundary conditions listed in cases 2 and 3 are more accurate. For instance, Table4.1 lists a comparison between the induced EMFs from the FEA and the ones from

45

the measurement, where Eend and Eact denote the induced EMF of the search coilsin the end region and in the active region, respectively, and a tilde over a symboldenotes the rms value of the field quantity symbolised by the symbol. It clearlyshows that in cases 2 and 3, the results of the EMF of the search coil in the endregion are very close to the measurements. A concrete and detailed analysis of theend-region magnetic field in the induction machine can be found in Publication I.

Table 4.1: Induced EMF of Search Coils.

Current (A) QuantityFinite Element Analysis

MeasurementCase 1 Case 2 Case 3

500Eend (V) 0.31 0.35 0.34 0.36

Eact (V) 3.23 3.24 3.24 3.58

832Eend (V) 0.52 0.58 0.57 0.61

Eact (V) 5.38 5.40 5.40 5.61

1000Eend (V) 0.63 0.70 0.69 0.73

Eact (V) 6.47 6.49 6.50 6.86

4.2 Stator End-Winding Inductance

The stator end-winding inductance of a 3-phase, 6-pole, 1.25-MW squirrel-cage in-duction machine was calculated. The main parameters of the machine are listed inAppendix A as Machine II. The machine has a two-layer diamond winding consistingof form-wound multi-turn coils.

The way in which the model of the stator end winding was built is similar to theone described in Section 4.1. The multi-turn coils were modelled as single-turn solidcoils, and the laminated core was modelled as a homogeneous solid core with thesame magnetic and electrical anisotropy as in Section 4.1. Only one pole at one endof the machine was modelled, but the rotor was not included. The current in theend winding of the model was supplied from a current source. A linear B–H curvewas used in the ferromagnetic materials, as explained in Section 4.1 in detail. Thegoverning equation of the FEA is (4.2), with the electric scalar potential V omitted,as explained in Section 4.1.

As yet, a unique definition of stator end-winding inductance has not been proposed,so different authors calculated it in different ways. For instance, de Weerdt andBelmans (1995) took account of the coupling between the stator and the rotor intheir calculation, but most authors, such as Taieb Brahimi et al. (1993) and Arshadet al. (2005), did not consider the coupling. In this FEA, the above coupling between

46

the stator and the rotor was not covered. The stator end-winding inductance perphase was calculated because it could be used in corresponding 2-D coupled field–circuit FEAs as a parameter of the following circuit equation:

ustr = R stristr +dψstr

act

dt+ Lstr

σ,enddistr

dt(4.6)

where ustr, istr, and R str denote the voltage, the current, and the resistance of thestator phase winding, respectively, Lstr

σ,end the stator end-winding inductance perphase, and ψstr

act the flux linkage of the stator phase winding in the active region.

The calculation of Lstrσ,end was based on the magnetic energy associated with the stator

end winding — in other words, both the self-inductance of each end connection andthe mutual inductance among different end connections were included. Under thetime-harmonic analysis, the time average of the magnetic energy over one periodwas adopted. The following equation relating to the magnetic energy should hold:

Wm,tot = W srcm,act + W src

m,end = W regm,act + W reg

m,end (4.7)

where a bar over a symbol denotes the time average of the field quantity symbolisedby the symbol, Wm,tot the total magnetic energy, W src

m,act and W srcm,end the magnetic

energy associated with the coil sides5 and the magnetic energy associated with theend connections, respectively, and W reg

m,act and W regm,end the magnetic energy in the

active region and the magnetic energy in the end region, respectively. In addition,the following relations should hold:

W srcm,act ∝ l strFe (4.8)

W srcm,end = constant (4.9)

where l strFe denotes the stator core length. Among these quantities, W srcm,end was related

to L strσ,end, but it was not possible to calculate W src

m,end directly. As a result, thecalculation of W src

m,end had to depend on (4.7) — in other words, Wm,tot and W srcm,act

had to be calculated first.

Wm,tot was easy to calculate whereas the calculation of W srcm,act should be considered

carefully. W srcm,act did not equal W reg

m,act by reason of the effect of the stator end-windingleakage. Therefore, in the FEA, the active region of the model was evenly dividedinto 40 10-mm-thick slices perpendicular to the axial direction, as shown in Figure4.3. Because the middle portion of the active region was supposed not to be affectedby the end-winding leakage, the magnetic energy in each of those slices lying in themiddle portion should be equal. In other words, the magnetic energy in those sliceswas only associated with the coil sides. Therefore, the time average of the magneticenergy calculated in any of those slices times 40 equalled W src

m,act. With the known

5In this compendium, the definition of the term“coil sides” is from reference (Gross, 2007), andthe term “coil sides” means the portion of a coil that lies in the slots, as indicated in Figure 1.3.

47

(a) (b)

Figure 4.3: A general description of the slices used for calculating the magneticenergy. (a) The index of the slices in the active region of the model with the finiteelement mesh. (b) One of the slices in the active region. (Source: Publication II.)

W srcm,act and Wm,tot, W

srcm,end was calculated from (4.7) and then Lstr

σ,end was calculatedfrom

Lstrσ,end = 4p

2 W srcm,end

m(istr)2 (4.10)

where p denotes the number of pole pairs, and m the number of phases. Additionally,the frequency dependence of the end-winding inductance and that of the magneticenergy mentioned above were analysed, and the excitation frequency was variedfrom 0.05 Hz to 5 kHz discretely.

In sum, the calculations of the magnetic energy in those 40 slices verify that only theslices close to the end surface of the core are affected by the end-winding leakage,as shown in Figure 4.4. In terms of the calculations at different frequencies, it isfound that as the frequency goes up, Wm,tot decreases but W src

m,act remains constant.Therefore, there is a drop in W src

m,end, and correspondingly, Lstrσ,end also decreases. The

calculations prove that the following relations hold at any of the above frequencies:

W srcm,end = W reg

m,end (4.11)

W srcm,act = W reg

m,act. (4.12)

Moreover, because of the stator end-winding leakage, there is an increase in W regm,act as

the frequency increases, but both W srcm,end and W reg

m,end decrease. The drop in W regm,end

is more than the increase in W regm,act, which indicates that Wm,tot decreases as the

frequency goes up. It is speculated that the frequency dependence is related to theeddy currents in the end region as well as in the core ends. A concrete calculation ofthe end-winding inductance of the induction machine can be found in Publication II.

48

0 5 10 15 20 25 30 35 400.11

0.114

0.118

0.122

0.126

0.13

0.134

Index of Slice

Mag

netic

Ene

rgy

(J)

Figure 4.4: The time average of the magnetic energy at 50 Hz in each of the 40slices in the active region. (Source: Publication II.)

4.3 Eddy Currents inside End Shield and End Frame

The electric machine used in this FEA is the same as the one in Section 4.1, and itsmain parameters can be found in Appendix A as Machine I. The eddy currents aswell as the eddy-current loss inside the end shield and the end frame were analysed.The rotor was removed so that the machine could be tested in the laboratory.

The model of the stator end winding was built in terms of the measurements. How-ever, in order that the eddy currents inside the end shield and the end frame could beanalysed accurately with a dense finite element mesh, the model of the end windingwas simplified — in other words, each end connection in the model was made up of12 connective segments as shown in Figure 4.5. The multi-turn coils were modelledas single-turn solid coils, and the laminated core was modelled as a homogeneoussolid core with the same magnetic and electrical anisotropy as in Section 4.1. Onlyone end of the machine was modelled, but the rotor was not included. The finiteelement mesh of the end region is similar to the one shown in Figure 4.1(a).

The current in the end winding of the model was supplied from a current source.On account of the extremely long computation time, a linear B–H curve was usedin the ferromagnetic materials, as discussed in Section 4.1. The governing equationof the FEA is (4.2), with the electric scalar potential V dropped because of thelimited computational resources, as explained in Section 4.1. Since in 3-D eddy-current problems, both the electric and the magnetic field must be described inthe conductive region (Bıro and Preis, 1990), the neglect of V may lead to aninaccurate estimation of the eddy-current loss. Therefore, a simple model of theinduction machine was built for analysing the effect of V on the eddy-current loss.The simple model is similar to the above model except that it included only oneend connection; hence, it was capable of being solved by the computer under theA–V formulation described by (3.11) and (3.12). The calculations showed that theeddy-current loss inside the end shield and the end frame was around 10% smaller

49

(a) (b)

Figure 4.5: A certain end connection of the model of a 3-phase, 4-pole, 2.24-MWsquirrel-cage induction machine. (a) The 12 connective segments. (b) The finiteelement mesh.

when σ·∇V was included.

As explained in Section 4.1, the end shield and the end frame were not capable ofbeing modelled because of the limited computational resources. They were morethan 10 mm thick, but the skin depth there was just around 1 mm at 50 Hz whenreluctivity 0.001ν0 was used. Therefore, the eddy currents could be regarded as sur-face currents flowing within the inner surface of those parts. In this time-harmonicFEA, those surface currents were modelled with the SIBC listed in (4.5). In fact,the SIBC is based on the relation between the tangential component of the electricfield and that of the magnetic field on the surface of the conductive region, and itallows only the surface of the region to be discretised (Gyselinck et al., 2009). Adetailed explanation of the SIBC and its application can be found in relevant booksand papers, such as references (Hoppe and Rahmat-Samii, 1995), (Jayasekera andCiric, 2007), and (Alotto et al., 2007).

Since the eddy currents were considered as surface currents, the corresponding eddy-current loss was calculated from

PFt =1

2

∫∫Γ

Re J sur·E∗ dS (4.13)

where PFt denotes the eddy-current loss, Jsur the surface current density, Γ thesurface of integration, S the surface area, and a star and “Re” beside a complexvector or a phasor denote the complex conjugate and the real part of the complexvector or the phasor, respectively.

In this analysis, how to validate the calculations of the eddy-current loss was aconsiderable problem. Though it was feasible to measure the total losses of the

50

induction machine, it was not possible to separate the loss inside the end shield andthe end frame from the total losses. Consequently, an indirect method of measuringthe temperature rise was adopted. In order for the measurement, a 1-mm-thicksteel sheet was rolled up, and heat insulation filler was put around it. The steelsheet was then put into the end frame in such a way that it was tightly attachedto the inner surface of the end frame. Owing to the small temperature differentialbetween the winding and the steel sheet, the thermal radiation was negligible. Thethermal conduction and the thermal convection were also omitted. As a result, thetemperature rise in the steel sheet was attributed to the eddy-current loss there.Four thermocouples were fixed to the steel sheet in different axial positions forrecording the change in temperature. The local dissipated power density in thesteel sheet was estimated from

pd = cstl ϱstl∆ϑ

∆t(4.14)

where pd denotes the dissipated power density, cstl the specific heat capacity ofsteel, ϱstl the mass density of steel, ∆ϑ the incremental temperature, and ∆t theincremental time.

In the real machine, there are two main parameters related to the eddy-currentloss inside the end shield and the end frame, and they are the minimum radialdistance between the end frame and the end winding, rmin, and the minimum axialdistance between the end shield and end winding, zmin, as indicated in Figure 4.6.From the viewpoint of electrical insulation, both of their minima can be around 7mm. In order for the analysis of their effect on the eddy-current loss, some 3-Dmodels were built. The above two parameters in every 3-D model were different,ranging from 7 mm to 40 mm. The eddy-current loss in each of the 3-D modelswas calculated with the same time-harmonic FEA as above. Additionally, the eddy-current loss was calculated from nonlinear time-discretised FEAs so that the effectof the magnetic nonlinearity in the end shield and the end frame could be analysed.

Figure 4.6: The minimum radial distance between the end frame and the endwinding, rmin, and the minimum axial distance between the end shield and the endwinding, zmin.

51

The nonlinear FEAs were carried out with simplified 2-D axisymmetric models forsaving the computation time, and the above two parameters in every 2-D modelwere also different. A nonlinear single-valued B–H curve was applied to the endshield and the end frame of the 2-D models. The corresponding eddy-current losswas calculated from

PφFt =

1

T1

T1∫0

∫∫Γ

σ

(∂Aφ

∂t

)2

dS dt (4.15)

where PφFt denotes the eddy-current loss per unit circumferential length, and T1 the

period of the fundamental frequency.

In sum, the use of the SIBC to model the eddy currents inside the end shield andthe end frame is feasible for large-sized induction machines. The measurement ofthe temperature rise is also a feasible solution to estimating the corresponding eddy-current loss. In terms of the 3-D time-harmonic FEAs, as shown in Figure 4.7, when

5 10 15 20 25 30 35 4050

100

150

200

250

300

350

Minimum Radial Distance (mm)

Edd

y−C

urre

nt L

oss

(W)

End shieldFrameEnd shield and frame

(a)

5 10 15 20 25 30 35 4050

100

150

200

250

300

350

Minimum Axial Distance (mm)

Edd

y−C

urre

nt L

oss

(W)

End shieldFrameEnd shield and frame

(b)

Figure 4.7: The variation in the eddy-current loss inside the end shield and theend frame of one end. (a) Only the minimum radial distance rmin is varied. (b)Only the minimum axial distance zmin is varied. (Source: Publication III.)

52

rmin and zmin are varied gradually from 7 mm to 40 mm, the eddy-current loss insidethose parts also changes gradually. Relatively high loss density arises inside the endframe near the connection between the nose portion and the involute portion of theend connections, as shown in Figure 4.8, but the total loss is small in comparisonwith other types of loss, e.g. the copper loss of the stator phase winding. In the 2-Dtime-discretised FEAs, on the basis of the fact that the reluctivity was supposed tobe 0.001ν0 in the linear case, the eddy-current loss calculated in the nonlinear caseunder any combination of rmin and zmin is around 10% larger than the loss calculatedin the corresponding linear case. A concrete and detailed analysis of the eddycurrents and the corresponding eddy-current loss can be found in Publication III.

0

50

100

150

200

250

300 (W/m )2

Figure 4.8: The distribution of the time average of the eddy-current loss densitywithin the inner surface of the end shield and the end frame at rmin = 20 mm andzmin = 20 mm. (Source: Publication III.)

Only the eddy currents induced by the stator end-winding leakage were consideredin the 3-D models of the squirrel-cage induction machine. The leakage flux comingfrom the rotor end cage was not covered since the rotor was not included in theFEAs. As a matter of fact, the position of the rotor end cage is far from theposition of the end shield and the end frame in comparison with the position ofthe stator end winding, so their effect on the eddy-current loss is probably not soimportant as the effect of the stator end winding under the steady-state operation.However, with the rotor rotating, the resultant leakage field coming from the statorend winding as well as the rotor end cage may cause a new distribution of the 3-Dend-region magnetic field; hence, there may be a change in the distribution of theeddy currents inside the end shield and the end frame as well.

53

5 Mechanical Analysis of End Region

This chapter outlines the mechanical FEA of the end region of two large-sized 3-phase squirrel-cage induction machines. The FEA covers two main aspects: ananalysis of the end-winding magnetic forces and an analysis of the end-windingmechanical vibrations. A detailed account of the above two aspects is not providedin this chapter, but it can be found in Publications IV and V.

5.1 End-Winding Magnetic Forces

The magnetic forces exerted on the stator end winding of a 3-phase, 6-pole, 1.25-MW squirrel-cage induction machine under the steady-state full-load were analysed.The main parameters of the machine are listed in Appendix A as Machine II.

The way in which the 3-D model of the stator end winding was built is similar tothe one described in Section 4.1. The multi-turn stator coils of the machine werereplaced by single-turn solid coils in the model. A careful representation of thegeometric shape of the end connections was of importance for the calculation of theend-winding forces, as pointed out by Lawrenson (1965), so each end connectionwas made up of 29 connective segments. In addition, in order that only one polecould be modelled, the number of rotor slots per pole should be an integer; hence,the number of rotor slots in the model was reduced from the original 86 to 84.According to Williamson and Ralph (1983), the rotor slot width should be changedaccordingly so that the rotor tooth magnetomotive force drop could be kept thesame. However, the rotor slot width was not changed since the calculation of theend-winding forces was the point of this analysis. Figure 5.1 shows part of the finiteelement mesh relating to the end region of the model.

The current in the end winding of the model was supplied from a current source.The magnetic nonlinearity of the ferromagnetic materials was not taken into accountowing to the extremely long computation time, as explained in Section 4.1. As aresult, a time-harmonic FEA was performed, in which the rotor of the model wassupposed to be pseudostationary. The governing equation of the FEA was (4.2).The electric scalar potential V was not solved, as explained in Section 4.1. In themodel, tensor conductivity sσ was used in the rotor, where s denotes the slip, so thefrequency of the stator field was used in both the stator and the rotor. Furthermore,in order for a simplification of the numerical computation, the eddy currents insidethe end shield and the end frame were omitted, so the boundary condition shownin (4.3) was applied to the inner surface of those parts.

The end-winding magnetic forces were classified as Lorentz forces. In the FEA ofthe 3-D model with 84 rotor slots, the steady-state forces on the end connections

54

Figure 5.1: Part of the finite element mesh relating to the end region of the modelof a 3-phase, 6-pole, 1.25-MW squirrel-cage induction machine.

of the three phase belts under a certain pole were somewhat different because thenumber of rotor slots per phase belt was not an integer. The time-varying end-winding forces on all the segments of the four end connections of a certain phasebelt were calculated from

Few =∫∫∫Ω

J ×B dV (5.1)

where Few denotes the end-winding force, Ω the domain of integration, and V thevolume. B was calculated from the FEA of the end-region magnetic field. The forcecomponents on a certain segment i were derived from (5.1). They are expressed asthe following functions of time:

Fi,r = Fi,r + Fi,r,∼ cos (2ωst+ ϑi,r0)

Fi,φ = Fi,φ + Fi,φ,∼ cos (2ωst+ ϑi,φ0)

Fi,z = Fi,z + Fi,z,∼ cos (2ωst+ ϑi,z0)

, i = 1, · · · , 29 (5.2)

where a hat over a symbol denotes the amplitude of the field quantity symbolisedby the symbol, i the index, Fi,r, Fi,φ, and Fi,z denote the r-, the φ-, and the z-component of the force on segment i, ωs the angular frequency of the stator field,Fi,r,∼, Fi,φ,∼, and Fi,z,∼ the double-frequency force components, and ϑi,r0, ϑi,φ0, andϑi,z0 the phase angles. Obviously, each force component consists of a constant anda double-frequency component.

Because the 29 connective segments of each end connection were not equal in volume,in order for a comparison of the force components on different segments of a certainend connection, the force density of a certain segment i was defined and calculatedas

fi =Fi

Vi, i = 1, · · · , 29 (5.3)

55

where fi denotes the force density of segment i, Fi the force on segment i, and Vi thevolume of segment i. The force density components of segment i are also expressedas the following functions of time:

fi,r =Fi,r

Vi+Fi,r,∼

Vicos (2ωst+ ϑi,r0)

fi,φ =Fi,

Vi+Fi,φ,∼

Vicos (2ωst+ ϑi,φ0)

fi,z =Fi,z

Vi+Fi,z,∼

Vicos (2ωst+ ϑi,z0)

, i = 1, · · · , 29 (5.4)

where fi,r, fi,φ, and fi,z denote the r-, the φ-, and the z-component of fi, respectively.

As a matter of fact, the most important six parameters in (5.4) are Fi,r/Vi, Fi,r,∼/Vi,

Fi,φ/Vi, Fi,φ,∼/Vi, Fi,z/Vi, and Fi,z,∼/Vi. These six parameters were calculated in allthe segments of the four end connections of the phase belt. The calculations indicatethat their values are basically larger in the knuckle portion of the end connectionsthan in the nose portion and the involute portion. The largest radial force densityappears in the knuckle portion lying in the inner layer of the end winding. Thedistribution of Fi,r/Vi and that of Fi,r,∼/Vi among the four end connections are quitesimilar, in comparison with the distribution of the other four parameters relating tothe circumferential and the axial force density. In addition, it is clearly found thatFi,r/Vi, Fi,φ/Vi, and Fi,z/Vi tend to spread all the end connections out in terms oftheir direction in different segments. A concrete and detailed analysis of the end-winding magnetic forces under the steady-state operation can mainly be found inPublication IV and partly in Publication V.

5.2 End-Winding Mechanical Vibrations

The end-winding mechanical vibrations caused by the steady-state end-windingforces in a 3-phase, 4-pole, 2.24-MW squirrel-cage induction machine were anal-ysed. The main parameters of the machine can be found in Appendix A as MachineI. The rotor was removed because it was not possible to test the induction machineunder full-load in the laboratory. The end connections of the multi-turn stator coilswere braced by an end-winding bracing system. The bracing system included blocks,tapes, and rings, which were made from epoxy resin-impregnated glass fibre. Theblocks were fixed between two neighbouring end connections for reducing their rela-tive motion mainly in the φ-direction. The tapes were capable of fastening two endconnections in the r-direction for restricting their relative motion mainly in the r-and the φ-direction. The rings surrounded all the end connections for restrictingtheir motion mainly in the r-direction. Furthermore, the stator, including the endwinding and its bracing system, had a global VPI process so that the end windingcould possess good mechanical strength and good resistance to moisture and chem-

56

icals (Sadarangani, 2000; Pyrhonen et al., 2008). From a mechanical point of view,the end winding and its bracing system constituted a single entity.

In general, the function of the end-winding bracing system of large-sized radialflux electric machines is to make the end connections withstand large forces dur-ing starting and short circuits and to reduce end-winding vibrations (Sadarangani,2000). It comprises non-conducting parts because of the electric field in the endregion (Klempner and Kerszenbaum, 2004). The bracing system is very stiff in ther-direction for minimising the vibration level. However, it is allowed to move axiallyto accommodate the axial thermal expansion of the stator winding (Klempner andKerszenbaum, 2004), because the forces resulting from the thermal expansion mayreach a high level (Lambrecht and Berger, 1983). Though the bracing system canbe made extremely stiff by a use of more bracing parts, such as blocks and rings,the cooling of the end region can be seriously affected.

The end-region magnetic field should be solved first before an analysis of the end-winding vibrations; hence, a complete model of the end region was required in orderfor the electromagnetic FEA. The complete model was called the electromagneticmodel in this FEA. The insulated multi-turn coils were replaced by uninsulatedsingle-turn solid coils in the model, so the insulation layers of the coils were notmodelled. The laminated core was modelled as a homogeneous solid core with thesame magnetic and electrical anisotropy as in Section 4.1. The end-winding bracingsystem was included in the electromagnetic model, but it was simplified on accountof its complexity. The mechanical effect of the simplified bracing system was similarto that of the real one, but the effect of the enamel cured during the VPI processwas not taken into consideration. Concerning the calculation of the vibrations, e.g.the calculation of a modal model, the mechanical model used was only a part ofthe electromagnetic model, which just included the end winding and its simplifiedbracing system.

The mechanical model was an LTI multi-DOF system, and Figure 5.2 shows part ofthe finite element mesh of the model. An undamped free-vibration FEA was com-pleted in order for relevant modal data, which was based on the following governingequation:

Mu+Ku= 0 (5.5)

where u denotes a column vector of the displacement. An impact hammer test, inwhich accelerance frequency response functions (FRFs) were measured, was com-pleted in order for the experimental modal model. Only a column of entries of theaccelerance FRF matrix was measured for the modal analysis according to He andFu (2001) and Ewins (2000). The experimental modal model was compared withthe calculated one derived from the FEA in order for possible model updating. Inaddition, in order that relevant correlated mode pairs (CMPs) could be identified,the modal assurance criterion (MAC) was calculated from the following expression

57

Figure 5.2: Part of the finite element mesh of the mechanical model of a 3-phase,4-pole, 2.24-MW squirrel-cage induction machine.

given in reference (Allemang, 2003):

MAC(ψ

calc,ψ

exp

)=

∣∣∣ψ Hexpψ

calc

∣∣∣2(ψ H

expψ

exp

) (ψ H

calcψ

calc

) (5.6)

where ψexp and ψcalc denote a column vector of the experimental mode shape anda column vector of the calculated mode shape, respectively, and “H” near a symboldenotes the Hermitian transpose of the matrix symbolised by the symbol. Actually,the experimental mode shape vector was a vector of complex numbers whereas thecalculated one was a vector of real numbers. After the comparison of the naturalfrequencies of the CMPs, according to Ewins (2000), the geometric structure of themodel proved reasonable, but the material properties had to be updated becausethe simplifications made to the mechanical model changed its mechanical structure.As a result, the main elastic properties of the materials, i.e. the Young’s modulusand the Poisson’s ratio, were updated accordingly until the calculated modal modelwas close to the experimental one.

Next, a time-harmonic electromagnetic FEA, based on the governing equation (4.2),was performed for calculating the end-region magnetic field. The electric scalar po-tential V was not solved because of the limited computational resources, as explainedin Section 4.1. In addition, a time-harmonic undamped forced-vibration FEA wascarried out, which was based on the following governing equation:

Mu+Ku= Fexc (5.7)

where Fexc denotes a column vector of the excitation forces. The excitation forceswere calculated from the previous electromagnetic FEA, and they were applied to thenodes of certain finite elements of the mechanical model by the software packagesince nodal elements were used in the mechanical FEA. Damping was omitted,

58

because the damping ratios derived from the impact hammer test were small andthe excitation frequency was much lower than the natural frequency of the mostexcitable mode derived from the impact hammer test.

The constant force component and the double-frequency force component discussedin Section 5.1 were dealt with individually in the forced-vibration FEA because ofthe LTI characteristics of the mechanical model. Corresponding measurement of

0.38 (ìm)

0

Rotatingmagnetic field

(a)

0.51 ( )ìm

0

Rotatingmagnetic field

Rotating deformation

(b)

Figure 5.3: The deformation of the stator end winding and its bracing system. (a)The static deformation caused by the constant force component. (b) The dynamicdeformation caused by the double-frequency force component. (Source: Publica-tion V.)

59

the operating deflection shape (ODS) was also performed. Moreover, in order foran analysis of the metal fatigue of the copper conductors of the end connections, acalculation of the von Mises stresses and an analysis of the Goodman relationshipwere completed, respectively.

In sum, in terms of the results, both the constant force component and the double-frequency force component are larger in the knuckle portion and part of the involuteportion of the end connections, but relatively strong deformation appears in thenose portion, as indicated by the ellipses in Figure 5.3. With the end-winding brac-ing system, the maximum amplitude of vibration is within 100 µm and the naturalfrequency of the most excitable mode, 315.6 Hz, is much higher than the excitationfrequency, 100 Hz. The calculations based on the von Mises stresses and the Good-man relationship indicate that the metal fatigue of the copper conductors caused bythe double-frequency force component is not an important issue. Furthermore, thefrequency-spectrum analysis of the time-varying displacement, based on the DFT ofthe displacement components recorded during the ODS measurement, shows that inaddition to the expected 100-Hz component, a strong 50-Hz component is presentin the frequency spectrum. There are two reasons for the 50-Hz component: oneis the interference coming from the electromagnetic sensitivity of the piezoelectricaccelerometers, and the other is the current in the connection wire of the accelerom-eters. A concrete and detailed analysis of the end-winding mechanical vibrationsunder the steady-state operation can be found in Publication V.

60

6 Conclusions and Discussions

As the last chapter of this compendium, this chapter summarises the main resultsand discusses the scientific significance of this piece of research. Furthermore, someissues relating to the accuracy of the research methods are discussed.

6.1 Main Results of Research

The main results obtained from the electromagnetic and the mechanical FEA of theend region of the above two large-sized 3-phase squirrel-cage induction machinesare divided into five aspects: the end-region magnetic field, the stator end-windinginductance, the eddy currents in the end region, the end-winding forces, and theend-winding vibrations.

First, a 3-D time-harmonic FEA of the end-region magnetic field is accurate enough.The magnetic nonlinearity caused by the saturation is not distinct in the end region.By comparison with the active region, the magnetic induction in the end region isquite weak, even near the knuckle portion of the end connections. The eddy currentsinside the end shield and the end frame can be modelled with the SIBC, but theyhave only a slight effect on the end-region magnetic field. In fact, applying the SIBCor a homogeneous Neumann boundary condition to the inner surface of those partscan lead to an accurate analysis of the end-region magnetic field, but the use of ahomogeneous Dirichlet boundary condition may cause some errors.

Second, the magnetic energy associated with the stator end winding can be separatedfrom the magnetic energy associated with the whole stator winding by a 3-D FEAof the magnetic field in the end region and in the active region, as the end-windingleakage just affects the magnetic field in the core ends. The stator end-windinginductance can be calculated from the corresponding magnetic energy. Moreover,the magnetic energy in the end region does not equal the magnetic energy associatedwith the stator end winding, and the former is a little smaller than the latter.

Third, the eddy currents inside the end shield and the end frame can be regardedas surface currents, and it is feasible to apply the SIBC to the inner surface ofthose parts in order for modelling of the eddy currents with a 3-D time-harmonicFEA. Even if those parts are very close to the end winding, the corresponding eddy-current loss is still small. Furthermore, the eddy-current loss calculated in a 3-Dtime-harmonic FEA is somewhat smaller than that in the real situation because ofthe neglect of the saturation.

Fourth, the steady-state end-winding forces calculated in a 3-D time-harmonic FEAconsist of a constant and a double-frequency component. The largest radial force

61

density appears in the knuckle portion of the end connections lying in the innerlayer of the end winding. In a certain phase belt, the end-winding forces on each ofthe end connections are mostly different.

Fifth, in spite of the complexity of the end region, the end-winding vibrations canbe analysed with a 3-D time-harmonic FEA based on an LTI mechanical modelin which a simplified end-winding bracing system is used. However, because ofthe simplifications, modal testing is required in order for relevant modal data andpossible model updating. With the end-winding bracing system, the maximumamplitude of vibration is quite small under the steady state. The natural frequencyof the most excitable mode of the end winding and its bracing system is also raisedto a high frequency avoiding resonances. Furthermore, the metal fatigue related tothe cyclic loading cannot appear in the copper conductors of the end connections.

In sum, most of the important aspects relating to the end region of the two inductionmachines are covered in this piece of research. Although the modelling conditionsused in the FEA are not exactly the same as the actual situation, the main resultsgiven above still provide a deep understanding of the end region, which is beneficialto the electromagnetic and mechanical design of the end region.

6.2 Scientific Significance of Research

This piece of research is mainly related to the design of large-sized radial flux electricmachines. Although the end region is normally not so important in the analysis ofelectric machines as the active region, its fast and reliable 3-D FEA is more effectivethan its conventional analytical analysis and quasi-3-D FEA in the design and theoptimisation of electric machines.

First, nowadays, a time-harmonic FEA is seldom used in the 2-D analysis of theactive region of radial flux electric machines because a time-discretised FEA is nolonger time-consuming on account of the development of computers. However, forthe 3-D analysis of the end region, a time-discretised FEA is still too time-consumingto carry out. As a result, a 3-D time-harmonic FEA, together with a use of suitableboundary conditions, can be regarded as an efficient solution to the end-region mag-netic field because it does not require much computation time but gives acceptableaccuracy.

Second, by the method of calculating the stator end-winding inductance proposedin this piece of research, the magnetic energy associated with the stator end windingcan be easily derived from a 3-D time-harmonic FEA based on a model with a fixedcore length, instead of using many models with different core lengths for carryingout linear extrapolation. The method is not restricted to a time-harmonic FEA;conversely, it can be extended to a time-discretised FEA.

62

Third, the eddy-current loss inside the end shield and the end frame is small, evenif the end winding is very close to those parts. However, the small loss does notmean that hot spots cannot appear because the distribution of the loss density isnon-uniform. As relatively high loss density appears inside the end frame near theconnection between the nose portion and the involute portion, the minimum radialdistance between the end winding and the end frame is an important parameter.From the viewpoint of optimisation, the end frame can be shifted farther fromthe end winding in order for elimination of potential hot spots. By contrast, theminimum axial distance between the end winding and the end shield is not importantsince the loss density there is much smaller than that in the end frame.

Fourth, modelling the multi-turn stator coils and the complex end-winding bracingsystem in a mechanical FEA is normally accompanied by correction of the materialproperties of the model, in particular its elastic properties. In addition, under thesteady-state operation, the knuckle portion experiences larger radial forces than theother portions, but relatively strong deformation and relatively large amplitude ofvibration appear in the nose portion. However, the steady-state vibration levels aresmall, so the end-winding vibrations can hardly affect the end-region magnetic field.In other words, the interaction between the end-region magnetic field and the end-winding vibrations is negligible. If a stiffer end-winding bracing system is needed,more bracing parts can probably be fixed to the nose portion. The knuckle portionand the lower part of the involution portion seem not to require many bracing parts,and fewer bracing parts there are also beneficial to the cooling of the end region.

In sum, the above scientific significance covers both the FEA and the design ofthe end region. Though only squirrel-cage induction machines are covered in thispiece of research, the above significance is also beneficial to other types of radial fluxelectric machines, such as wound rotor induction machines, wound field synchronousmachines, and PM synchronous machines.

6.3 Accuracy of Research Methods

The FEA of the end region is the main part of this piece of research, and thecorresponding measurement is a complement to the FEA. Neither the FEA nor themeasurement is absolutely perfect. The following is a summary of the issues relatingto the accuracy of the research methods.

1. Only the main parts in the end region were included in the 3-D models; forexample, the bearings were not included. In order that boundary conditionscould be applied conveniently in the model including the rotor, the rotorshaft was shortened, and it was completely inside the end region boundedby the end shield and the end frame. The insulated multi-turn stator coils

63

were replaced with uninsulated single-turn solid coils. The laminated core wasmodelled as a homogeneous solid core with magnetic and electrical anisotropy.The complex end-winding bracing system was also simplified.

2. As both the end region and at least part of the active region were required inthe 3-D models, the number of DOFs could be greatly reduced with the use ofprismatic edge elements in the active region, as shown in Figure 4.3(a). How-ever, tetrahedral nodal and edge elements had to be used in the end region.When antiperiodic boundary conditions were applied, extremely narrow orirregular spaces were often generated somewhere in the end region. Tetra-hedral elements in those spaces sometimes had an extremely low elementquality, so the coefficient matrix of the system of equations was sometimessingular. Therefore, attention should be paid to the element quality.

3. The 3-D FEA of the end region did not take account of the following fourmain factors: the saturation of the ferromagnetic materials; the hysteresisof the ferromagnetic materials; the rotation of the rotor, and the mechani-cal nonlinearity of the structure. The first three are more important thanthe last one. The saturation can lower the peak magnetic induction in thesaturated region and cause corresponding harmonics; the hysteresis can some-what increase the loss in the ferromagnetic materials, and the rotation of therotor can also cause higher harmonics, etc. Therefore, these three factorscan somewhat affect the calculations of the end-region magnetic field and theeddy-current loss, etc. The reason why they were not considered in this pieceof research results mainly from the limited computational resources and theextremely long computation time. The fourth factor is not important becausethe end-winding vibrations and the deformation were not large.

4. In the course of the modal testing of the 4-pole induction machine, the numberof layers measured in the z-direction was three, and the number of spotsmeasured in each layer in the φ-direction was eight. Actually, the threemeasured layers and the eight measured spots in each layer are not sufficientand are only able to show such mode shapes associated with lower naturalfrequencies roughly. More layers measured in the nose portion and morespots measured in each layer in the φ-direction will be advantageous to acomplete description of the mode shapes and to finding modes with highernatural frequencies.

5. During the ODS measurement of the end-winding, the time-varying magneticfield in the machine interfered with the piezoelectric accelerometers. Theinterference can be eliminated by a use of fibre-optic accelerometers, as inreferences (Humer et al., 2008) and (Shally et al., 2008). On the other hand,in the course of the modal testing in which a piezoelectric accelerometer wasalso used, since no magnetic field was present in the machine, the aboveinterference did not occur.

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A–1

Appendix A Main Parameters of Electric Machines

Table A.1: Main Parameters of Induction Machines I and II.

Machine I Machine IIParameter Value ValueRated power (MW) 2.24 1.25Rated voltage (V) 1000 690Rated frequency (Hz) 50 50Rated slip 0.37% 0.365%Full length of stator core (mm) 1000 800Full length of rotor core (mm) 1010 810Outer diameter of stator core (mm) 980 980Inner diameter of stator core (mm) 670 670Outer diameter of rotor core (mm) 662 663Number of phases 3 3Number of pole pairs 2 3Number of stator slots 72 72Number of rotor slots 86 86Number of layers of stator winding 2 2Number of turns in series of stator coil 3 3Number of parallel branches of stator winding 4 3Coil span of stator coil (stator slot pitches) 16 10Skewing of stator slots 0 0Skewing of rotor slots 0 0Connection of stator winding Delta Delta

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